Termination of the given ITRSProblem could not be shown:



↳ ITRS
  ↳ ITRStoQTRSProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

eval(x, y) → Cond_eval(&&(>@z(x, 0@z), >=@z(y, x)), x, y)
Cond_eval(TRUE, x, y) → eval(-@z(x, 1@z), y)
eval(x, y) → Cond_eval1(&&(>@z(x, 0@z), >@z(x, y)), x, y)
Cond_eval1(TRUE, x, y) → eval(y, y)

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval1(TRUE, 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:

eval(x, y) → Cond_eval(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
Cond_eval(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
Cond_eval1(true, x, y) → eval(y, 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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:

EVAL(x, y) → COND_EVAL(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
EVAL(x, y) → AND(greater_int(x, pos(0)), greatereq_int(y, x))
EVAL(x, y) → GREATER_INT(x, pos(0))
EVAL(x, y) → GREATEREQ_INT(y, x)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
COND_EVAL(true, x, y) → MINUS_INT(x, pos(s(0)))
EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
EVAL(x, y) → AND(greater_int(x, pos(0)), greater_int(x, y))
EVAL(x, y) → GREATER_INT(x, y)
COND_EVAL1(true, x, y) → EVAL(y, y)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
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:

eval(x, y) → Cond_eval(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
Cond_eval(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
Cond_eval1(true, x, y) → eval(y, 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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:

EVAL(x, y) → COND_EVAL(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
EVAL(x, y) → AND(greater_int(x, pos(0)), greatereq_int(y, x))
EVAL(x, y) → GREATER_INT(x, pos(0))
EVAL(x, y) → GREATEREQ_INT(y, x)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
COND_EVAL(true, x, y) → MINUS_INT(x, pos(s(0)))
EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
EVAL(x, y) → AND(greater_int(x, pos(0)), greater_int(x, y))
EVAL(x, y) → GREATER_INT(x, y)
COND_EVAL1(true, x, y) → EVAL(y, y)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
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:

eval(x, y) → Cond_eval(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
Cond_eval(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
Cond_eval1(true, x, y) → eval(y, 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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 10 less nodes.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

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

The TRS R consists of the following rules:

eval(x, y) → Cond_eval(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
Cond_eval(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
Cond_eval1(true, x, y) → eval(y, 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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].

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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: 



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

eval(x, y) → Cond_eval(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
Cond_eval(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
Cond_eval1(true, x, y) → eval(y, 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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].

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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)

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: 



↳ 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:

eval(x, y) → Cond_eval(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
Cond_eval(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
Cond_eval1(true, x, y) → eval(y, 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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
              ↳ 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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].

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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:

eval(x, y) → Cond_eval(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
Cond_eval(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
Cond_eval1(true, x, y) → eval(y, 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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
              ↳ 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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].

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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))

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(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:

eval(x, y) → Cond_eval(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
Cond_eval(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
Cond_eval1(true, x, y) → eval(y, 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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
              ↳ 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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].

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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))

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

The following dependency pairs can be deleted:

GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ 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:

eval(x, y) → Cond_eval(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
Cond_eval(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
Cond_eval1(true, x, y) → eval(y, 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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
              ↳ 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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].

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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))

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

The following dependency pairs can be deleted:

GREATER_INT(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:

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(x, y) → COND_EVAL(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
COND_EVAL1(true, x, y) → EVAL(y, y)

The TRS R consists of the following rules:

eval(x, y) → Cond_eval(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
Cond_eval(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
Cond_eval1(true, x, y) → eval(y, 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:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof

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

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(x, y) → COND_EVAL(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
COND_EVAL1(true, x, y) → EVAL(y, y)

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
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))
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)
greater_int(pos(0), neg(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)), 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))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, 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].

eval(x0, x1)
Cond_eval(true, x0, x1)
Cond_eval1(true, x0, x1)



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing

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

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(x, y) → COND_EVAL(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y)
EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
COND_EVAL1(true, x, y) → EVAL(y, y)

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
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))
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)
greater_int(pos(0), neg(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)), 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))

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.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
Positions in right side of the pair: The new variable was added to all pairs as a new argument[CONREM].

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                          ↳ QDP
                        ↳ RemovalProof
                        ↳ Narrowing

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

COND_EVAL(true, x, y, x_removed) → EVAL(minus_int(x, x_removed), y, x_removed)
EVAL(x, y, x_removed) → COND_EVAL(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y, x_removed)
EVAL(x, y, x_removed) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y, x_removed)
COND_EVAL1(true, x, y, x_removed) → EVAL(y, y, x_removed)

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
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))
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)
greater_int(pos(0), neg(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)), 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))

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.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
Positions in right side of the pair: The new variable was added to all pairs as a new argument[CONREM].

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                          ↳ QDP
                        ↳ Narrowing

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

COND_EVAL(true, x, y, x_removed) → EVAL(minus_int(x, x_removed), y, x_removed)
EVAL(x, y, x_removed) → COND_EVAL(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y, x_removed)
EVAL(x, y, x_removed) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y, x_removed)
COND_EVAL1(true, x, y, x_removed) → EVAL(y, y, x_removed)

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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(x, y) → COND_EVAL(and(greater_int(x, pos(0)), greatereq_int(y, x)), x, y) at position [0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof

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

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
EVAL(neg(0), neg(s(x0))) → COND_EVAL(and(greater_int(neg(0), pos(0)), false), neg(0), neg(s(x0)))
EVAL(pos(0), pos(x0)) → COND_EVAL(and(greater_int(pos(0), pos(0)), true), pos(0), pos(x0))
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), neg(0)) → COND_EVAL(and(greater_int(pos(0), pos(0)), true), pos(0), neg(0))
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(and(greater_int(pos(s(x0)), pos(0)), false), pos(s(x0)), pos(0))
EVAL(pos(0), neg(s(x0))) → COND_EVAL(and(greater_int(pos(0), pos(0)), false), pos(0), neg(s(x0)))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(greater_int(neg(s(x1)), pos(0)), greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x1)), neg(x0)) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), false), pos(s(x1)), neg(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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 2 less nodes.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting

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

EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(pos(0), pos(x0)) → COND_EVAL(and(greater_int(pos(0), pos(0)), true), pos(0), pos(x0))
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), neg(0)) → COND_EVAL(and(greater_int(pos(0), pos(0)), true), pos(0), neg(0))
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(and(greater_int(pos(s(x0)), pos(0)), false), pos(s(x0)), pos(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(greater_int(neg(s(x1)), pos(0)), greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x1)), neg(x0)) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), false), pos(s(x1)), neg(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(0), pos(x0)) → COND_EVAL(and(greater_int(pos(0), pos(0)), true), pos(0), pos(x0)) at position [0,0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof

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

EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), neg(0)) → COND_EVAL(and(greater_int(pos(0), pos(0)), true), pos(0), neg(0))
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(and(greater_int(pos(s(x0)), pos(0)), false), pos(s(x0)), pos(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(greater_int(neg(s(x1)), pos(0)), greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x1)), neg(x0)) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), false), pos(s(x1)), neg(x0))
EVAL(pos(0), pos(x0)) → COND_EVAL(and(false, true), pos(0), pos(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting

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

COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), neg(0)) → COND_EVAL(and(greater_int(pos(0), pos(0)), true), pos(0), neg(0))
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(and(greater_int(pos(s(x0)), pos(0)), false), pos(s(x0)), pos(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(greater_int(neg(s(x1)), pos(0)), greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x1)), neg(x0)) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), false), pos(s(x1)), neg(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(0), neg(0)) → COND_EVAL(and(greater_int(pos(0), pos(0)), true), pos(0), neg(0)) at position [0,0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof

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

COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(and(greater_int(pos(s(x0)), pos(0)), false), pos(s(x0)), pos(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(greater_int(neg(s(x1)), pos(0)), greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x1)), neg(x0)) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), false), pos(s(x1)), neg(x0))
EVAL(pos(0), neg(0)) → COND_EVAL(and(false, true), 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
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))
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)
greater_int(pos(0), neg(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)), 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))

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
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting

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

EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(and(greater_int(pos(s(x0)), pos(0)), false), pos(s(x0)), pos(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(greater_int(neg(s(x1)), pos(0)), greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x1)), neg(x0)) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), false), pos(s(x1)), neg(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(s(x0)), pos(0)) → COND_EVAL(and(greater_int(pos(s(x0)), pos(0)), false), pos(s(x0)), pos(0)) at position [0,0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof

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

EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(greater_int(neg(s(x1)), pos(0)), greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x1)), neg(x0)) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), false), pos(s(x1)), neg(x0))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(and(true, false), pos(s(x0)), 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
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))
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)
greater_int(pos(0), neg(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)), 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))

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
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting

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

COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(greater_int(neg(s(x1)), pos(0)), greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x1)), neg(x0)) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), false), pos(s(x1)), neg(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0))) at position [0,0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting

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

COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(greater_int(neg(s(x1)), pos(0)), greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x1)), neg(x0)) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), false), pos(s(x1)), neg(x0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), 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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(greater_int(neg(s(x1)), pos(0)), greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0))) at position [0,0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting

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

COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), neg(x0)) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), false), pos(s(x1)), neg(x0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), 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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(s(x1)), neg(x0)) → COND_EVAL(and(greater_int(pos(s(x1)), pos(0)), false), pos(s(x1)), neg(x0)) at position [0,0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof

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

COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x1)), neg(x0)) → COND_EVAL(and(true, false), pos(s(x1)), neg(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing

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

EVAL(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), 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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(x, y) → COND_EVAL1(and(greater_int(x, pos(0)), greater_int(x, y)), x, y) at position [0] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), neg(0))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), neg(s(x1)))
EVAL(neg(0), neg(0)) → COND_EVAL1(and(greater_int(neg(0), pos(0)), false), neg(0), neg(0))
EVAL(neg(s(x0)), neg(0)) → COND_EVAL1(and(greater_int(neg(s(x0)), pos(0)), false), neg(s(x0)), neg(0))
EVAL(pos(0), neg(0)) → COND_EVAL1(and(greater_int(pos(0), pos(0)), false), pos(0), neg(0))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(pos(0), pos(0)) → COND_EVAL1(and(greater_int(pos(0), pos(0)), false), pos(0), pos(0))
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(neg(0), pos(s(x0))) → COND_EVAL1(and(greater_int(neg(0), pos(0)), false), neg(0), pos(s(x0)))
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(s(x0)), pos(0)) → COND_EVAL1(and(greater_int(neg(s(x0)), pos(0)), false), neg(s(x0)), pos(0))
EVAL(pos(0), neg(s(x0))) → COND_EVAL1(and(greater_int(pos(0), pos(0)), true), pos(0), neg(s(x0)))
EVAL(pos(0), pos(s(x0))) → COND_EVAL1(and(greater_int(pos(0), pos(0)), false), pos(0), pos(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), pos(0))
EVAL(neg(s(x0)), pos(s(x1))) → COND_EVAL1(and(greater_int(neg(s(x0)), pos(0)), false), neg(s(x0)), pos(s(x1)))
EVAL(neg(0), pos(0)) → COND_EVAL1(and(greater_int(neg(0), pos(0)), false), neg(0), pos(0))
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(greater_int(neg(s(x0)), pos(0)), greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(greater_int(neg(0), pos(0)), true), neg(0), neg(s(x0)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof

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

COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), neg(0))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), neg(s(x1)))
EVAL(neg(0), neg(0)) → COND_EVAL1(and(greater_int(neg(0), pos(0)), false), neg(0), neg(0))
EVAL(neg(s(x0)), neg(0)) → COND_EVAL1(and(greater_int(neg(s(x0)), pos(0)), false), neg(s(x0)), neg(0))
EVAL(pos(0), neg(0)) → COND_EVAL1(and(greater_int(pos(0), pos(0)), false), pos(0), neg(0))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(pos(0), pos(0)) → COND_EVAL1(and(greater_int(pos(0), pos(0)), false), pos(0), pos(0))
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(neg(0), pos(s(x0))) → COND_EVAL1(and(greater_int(neg(0), pos(0)), false), neg(0), pos(s(x0)))
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(s(x0)), pos(0)) → COND_EVAL1(and(greater_int(neg(s(x0)), pos(0)), false), neg(s(x0)), pos(0))
EVAL(pos(0), neg(s(x0))) → COND_EVAL1(and(greater_int(pos(0), pos(0)), true), pos(0), neg(s(x0)))
EVAL(pos(0), pos(s(x0))) → COND_EVAL1(and(greater_int(pos(0), pos(0)), false), pos(0), pos(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), pos(0))
EVAL(neg(s(x0)), pos(s(x1))) → COND_EVAL1(and(greater_int(neg(s(x0)), pos(0)), false), neg(s(x0)), pos(s(x1)))
EVAL(neg(0), pos(0)) → COND_EVAL1(and(greater_int(neg(0), pos(0)), false), neg(0), pos(0))
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(greater_int(neg(s(x0)), pos(0)), greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(greater_int(neg(0), pos(0)), true), 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
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))
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)
greater_int(pos(0), neg(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)), 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))

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 9 less nodes.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting

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

EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), neg(0))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(greater_int(neg(s(x0)), pos(0)), greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), neg(s(x1)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL1(and(greater_int(pos(0), pos(0)), true), pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), pos(0))
EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(greater_int(neg(0), pos(0)), true), 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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(s(x0)), neg(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), neg(0)) at position [0,0] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(true, true), pos(s(x0)), neg(0))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting

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

EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(greater_int(neg(s(x0)), pos(0)), greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), neg(s(x1)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL1(and(greater_int(pos(0), pos(0)), true), pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), pos(0))
EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(greater_int(neg(0), pos(0)), true), neg(0), neg(s(x0)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(true, true), pos(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1))) at position [0,0] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting

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

EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(greater_int(neg(s(x0)), pos(0)), greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), neg(s(x1)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL1(and(greater_int(pos(0), pos(0)), true), pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), pos(0))
EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(greater_int(neg(0), pos(0)), true), neg(0), neg(s(x0)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(true, true), pos(s(x0)), neg(0))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(greater_int(neg(s(x0)), pos(0)), greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1))) at position [0,0] we obtained the following new rules [LPAR04]:

EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting

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

EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), neg(s(x1)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL1(and(greater_int(pos(0), pos(0)), true), pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), pos(0))
EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(greater_int(neg(0), pos(0)), true), neg(0), neg(s(x0)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(true, true), pos(s(x0)), neg(0))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), neg(s(x1))) at position [0,0] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(true, true), pos(s(x0)), neg(s(x1)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting

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

EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(0), neg(s(x0))) → COND_EVAL1(and(greater_int(pos(0), pos(0)), true), pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), pos(0))
EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(greater_int(neg(0), pos(0)), true), neg(0), neg(s(x0)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(true, true), pos(s(x0)), neg(0))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(true, true), pos(s(x0)), neg(s(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(0), neg(s(x0))) → COND_EVAL1(and(greater_int(pos(0), pos(0)), true), pos(0), neg(s(x0))) at position [0,0] we obtained the following new rules [LPAR04]:

EVAL(pos(0), neg(s(x0))) → COND_EVAL1(and(false, true), pos(0), neg(s(x0)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof

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

EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), pos(0))
EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(greater_int(neg(0), pos(0)), true), neg(0), neg(s(x0)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(true, true), pos(s(x0)), neg(0))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(true, true), pos(s(x0)), neg(s(x1)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL1(and(false, true), pos(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting

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

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), pos(0))
EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(greater_int(neg(0), pos(0)), true), neg(0), neg(s(x0)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(true, true), pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(true, true), pos(s(x0)), neg(s(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(s(x0)), pos(0)) → COND_EVAL1(and(greater_int(pos(s(x0)), pos(0)), true), pos(s(x0)), pos(0)) at position [0,0] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(true, true), pos(s(x0)), pos(0))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting

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

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(greater_int(neg(0), pos(0)), true), neg(0), neg(s(x0)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(true, true), pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(true, true), pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(true, true), pos(s(x0)), 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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(neg(0), neg(s(x0))) → COND_EVAL1(and(greater_int(neg(0), pos(0)), true), neg(0), neg(s(x0))) at position [0,0] we obtained the following new rules [LPAR04]:

EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(false, true), neg(0), neg(s(x0)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof

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

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(true, true), pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(true, true), pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(true, true), pos(s(x0)), pos(0))
EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(false, true), 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
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))
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)
greater_int(pos(0), neg(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)), 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))

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
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting

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

EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(true, true), pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(true, true), pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(true, true), pos(s(x0)), 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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(s(x0)), neg(0)) → COND_EVAL1(and(true, true), pos(s(x0)), neg(0)) at position [0] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting

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

EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(true, true), pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(true, true), pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(true, true), pos(s(x0)), neg(s(x1))) at position [0] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting

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

EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(true, true), pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(s(x0)), pos(0)) → COND_EVAL1(and(true, true), pos(s(x0)), pos(0)) at position [0] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing

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

EVAL(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), 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
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))
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)
greater_int(pos(0), neg(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)), 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))

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(pos(s(x0)), y1) → COND_EVAL(and(true, greatereq_int(y1, pos(s(x0)))), pos(s(x0)), y1) at position [0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof

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

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))
EVAL(pos(s(x1)), neg(x0)) → COND_EVAL(and(true, false), pos(s(x1)), neg(x0))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(and(true, false), pos(s(x0)), 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
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))
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)
greater_int(pos(0), neg(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)), 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))

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 2 less nodes.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof

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

EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), 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
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))
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)
greater_int(pos(0), neg(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)), 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))

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.
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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing

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

EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
By narrowing [LPAR04] the rule EVAL(neg(0), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(0))), neg(0), y1) at position [0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof

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

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))
EVAL(neg(0), neg(0)) → COND_EVAL(and(false, true), neg(0), neg(0))
EVAL(neg(0), neg(s(x0))) → COND_EVAL(and(false, false), neg(0), neg(s(x0)))
EVAL(neg(0), pos(x0)) → COND_EVAL(and(false, true), neg(0), pos(x0))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing

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

EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
By narrowing [LPAR04] the rule EVAL(neg(s(x0)), y1) → COND_EVAL(and(false, greatereq_int(y1, neg(s(x0)))), neg(s(x0)), y1) at position [0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof

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

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))
EVAL(neg(s(y0)), neg(0)) → COND_EVAL(and(false, true), neg(s(y0)), neg(0))
EVAL(neg(s(y0)), pos(x0)) → COND_EVAL(and(false, true), neg(s(y0)), pos(x0))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof

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

EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing

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

EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
By narrowing [LPAR04] the rule EVAL(pos(0), y1) → COND_EVAL(and(false, greatereq_int(y1, pos(0))), pos(0), y1) at position [0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof

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

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))
EVAL(pos(0), pos(x0)) → COND_EVAL(and(false, true), pos(0), pos(x0))
EVAL(pos(0), neg(s(x0))) → COND_EVAL(and(false, false), pos(0), neg(s(x0)))
EVAL(pos(0), neg(0)) → COND_EVAL(and(false, true), pos(0), neg(0))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof

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

EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing

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

EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0))
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
By narrowing [LPAR04] the rule EVAL(neg(x1), pos(x0)) → COND_EVAL(and(greater_int(neg(x1), pos(0)), true), neg(x1), pos(x0)) at position [0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof

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

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))
EVAL(neg(s(x0)), pos(y1)) → COND_EVAL(and(false, true), neg(s(x0)), pos(y1))
EVAL(neg(0), pos(y1)) → COND_EVAL(and(false, true), neg(0), pos(y1))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing

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

EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0))
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
By narrowing [LPAR04] the rule EVAL(neg(x0), neg(0)) → COND_EVAL(and(greater_int(neg(x0), pos(0)), true), neg(x0), neg(0)) at position [0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof

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

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))
EVAL(neg(s(x0)), neg(0)) → COND_EVAL(and(false, true), neg(s(x0)), neg(0))
EVAL(neg(0), neg(0)) → COND_EVAL(and(false, true), neg(0), neg(0))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
By narrowing [LPAR04] the rule COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), y) at position [0] we obtained the following new rules [LPAR04]:

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))
COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), y1)
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greater_int(neg(0), pos(0)) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(neg(0), pos(s(y))) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_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.
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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))
COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), y1)
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)

The TRS R consists of the following rules:

minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(neg(0), pos(0)) → false
greater_int(neg(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.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))
COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), y1)
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)

The TRS R consists of the following rules:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(neg(s(x0)), y1) → COND_EVAL1(and(false, greater_int(neg(s(x0)), y1)), neg(s(x0)), y1) at position [0] we obtained the following new rules [LPAR04]:

EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(neg(s(x0)), neg(0)) → COND_EVAL1(and(false, false), neg(s(x0)), neg(0))
EVAL(neg(s(x0)), pos(s(x1))) → COND_EVAL1(and(false, false), neg(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), pos(0)) → COND_EVAL1(and(false, false), neg(s(x0)), pos(0))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))
COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), y1)
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(0)) → COND_EVAL1(and(false, false), neg(s(x0)), neg(0))
EVAL(neg(s(x0)), pos(s(x1))) → COND_EVAL1(and(false, false), neg(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), pos(0)) → COND_EVAL1(and(false, false), neg(s(x0)), 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))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(neg(0), pos(0)) → false
greater_int(neg(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)))
plus_nat(0, x0)
plus_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
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof

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

COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), y1)
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), 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))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(neg(0), pos(0)) → false
greater_int(neg(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)))
plus_nat(0, x0)
plus_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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing

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

COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), y1)
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(neg(0), y1) → COND_EVAL1(and(false, greater_int(neg(0), y1)), neg(0), y1) at position [0] we obtained the following new rules [LPAR04]:

EVAL(neg(0), pos(0)) → COND_EVAL1(and(false, false), neg(0), pos(0))
EVAL(neg(0), pos(s(x0))) → COND_EVAL1(and(false, false), neg(0), pos(s(x0)))
EVAL(neg(0), neg(0)) → COND_EVAL1(and(false, false), neg(0), neg(0))
EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(false, true), neg(0), neg(s(x0)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof

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

COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), y1)
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))
EVAL(neg(0), pos(0)) → COND_EVAL1(and(false, false), neg(0), pos(0))
EVAL(neg(0), pos(s(x0))) → COND_EVAL1(and(false, false), neg(0), pos(s(x0)))
EVAL(neg(0), neg(0)) → COND_EVAL1(and(false, false), neg(0), neg(0))
EVAL(neg(0), neg(s(x0))) → COND_EVAL1(and(false, true), neg(0), neg(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_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
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), y1)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), y1)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(pos(0), y1) → COND_EVAL1(and(false, greater_int(pos(0), y1)), pos(0), y1) at position [0] we obtained the following new rules [LPAR04]:

EVAL(pos(0), neg(s(x0))) → COND_EVAL1(and(false, true), pos(0), neg(s(x0)))
EVAL(pos(0), neg(0)) → COND_EVAL1(and(false, false), pos(0), neg(0))
EVAL(pos(0), pos(0)) → COND_EVAL1(and(false, false), pos(0), pos(0))
EVAL(pos(0), pos(s(x0))) → COND_EVAL1(and(false, false), pos(0), pos(s(x0)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), y1)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))
EVAL(pos(0), neg(s(x0))) → COND_EVAL1(and(false, true), pos(0), neg(s(x0)))
EVAL(pos(0), neg(0)) → COND_EVAL1(and(false, false), pos(0), neg(0))
EVAL(pos(0), pos(0)) → COND_EVAL1(and(false, false), pos(0), pos(0))
EVAL(pos(0), pos(s(x0))) → COND_EVAL1(and(false, false), pos(0), pos(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_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
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof

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

COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), y1)
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation

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

COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), y1)
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(true, pos(s(x0)), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(true, pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(s(z0), s(0)), pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_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
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting

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

COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(s(z0), s(0)), pos(s(z1)))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1)
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(pos(s(x0)), y1) → COND_EVAL1(and(true, greater_int(pos(s(x0)), y1)), pos(s(x0)), y1) at position [0] we obtained the following new rules [LPAR04]:

EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(true, true), pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(true, true), pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(true, true), pos(s(x0)), neg(0))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(z1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL1(and(true, true), pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL1(and(true, true), pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL1(and(true, true), pos(s(x0)), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_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
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof

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

COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(z1)))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing

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

COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(z1)))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(y1)))
COND_EVAL(true, pos(s(0)), pos(s(y1))) → EVAL(pos(0), pos(s(y1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_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
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                                ↳ UsableRulesProof

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

COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(y1)))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                                    ↳ QReductionProof

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

COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(y1)))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_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_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
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                                        ↳ Instantiation

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

COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(y1)))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL1(true, x, y) → EVAL(y, y)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL1(true, x, y) → EVAL(y, y) we obtained the following new rules [LPAR04]:

COND_EVAL1(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(z1)), neg(s(z1)))
COND_EVAL1(true, pos(s(z0)), pos(s(z1))) → EVAL(pos(s(z1)), pos(s(z1)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                                        ↳ Instantiation
                                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof

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

COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(y1)))
EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
COND_EVAL1(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(z1)), neg(s(z1)))
COND_EVAL1(true, pos(s(z0)), pos(s(z1))) → EVAL(pos(s(z1)), pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_nat(s(x0), x1)

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

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                                        ↳ Instantiation
                                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                                              ↳ AND
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                  ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                                ↳ QDP

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

COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
COND_EVAL1(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(z1)), neg(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_nat(s(x0), 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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                                        ↳ Instantiation
                                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                                              ↳ AND
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                  ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                                                                                      ↳ Instantiation
                                                                                                                                                                                                                                                                                                ↳ QDP

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

COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), y1)
EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
COND_EVAL1(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(z1)), neg(s(z1)))

The TRS R consists of the following rules:

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

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

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

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                                        ↳ Instantiation
                                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                                              ↳ AND
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                  ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                                                                                      ↳ Instantiation
                                                                                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                                                                                          ↳ Rewriting
                                                                                                                                                                                                                                                                                                ↳ QDP

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

EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
COND_EVAL1(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(z1)), neg(s(z1)))
COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(plus_nat(s(z0), s(0))), neg(s(z1)))

The TRS R consists of the following rules:

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

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

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

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                                        ↳ Instantiation
                                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                                              ↳ AND
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                  ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                                                                                      ↳ Instantiation
                                                                                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                                                                                          ↳ Rewriting
                                                                                                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                                                                                                ↳ QDP

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

EVAL(neg(s(x1)), neg(s(x0))) → COND_EVAL(and(false, greatereq_int(neg(x0), neg(x1))), neg(s(x1)), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL1(and(false, greater_int(neg(x0), neg(x1))), neg(s(x0)), neg(s(x1)))
COND_EVAL1(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(z1)), neg(s(z1)))
COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(plus_nat(z0, s(0)))), neg(s(z1)))

The TRS R consists of the following rules:

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

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

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

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                                        ↳ Instantiation
                                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                                              ↳ AND
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                  ↳ UsableRulesProof

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(y1)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL1(true, pos(s(z0)), pos(s(z1))) → EVAL(pos(s(z1)), pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
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)))
plus_nat(0, x0)
plus_nat(s(x0), 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
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                                        ↳ Instantiation
                                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                                              ↳ AND
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                  ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                                                                                      ↳ QReductionProof

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(y1)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL1(true, pos(s(z0)), pos(s(z1))) → EVAL(pos(s(z1)), pos(s(z1)))

The TRS R consists of the following rules:

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

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

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

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                                        ↳ Instantiation
                                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                                              ↳ AND
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                  ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                                                                                      ↳ QReductionProof
                                                                                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                                                                                          ↳ ForwardInstantiation

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

EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0)))
COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(y1)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL1(true, pos(s(z0)), pos(s(z1))) → EVAL(pos(s(z1)), pos(s(z1)))

The TRS R consists of the following rules:

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

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)))

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule EVAL(pos(s(x1)), pos(s(x0))) → COND_EVAL(and(true, greatereq_int(pos(x0), pos(x1))), pos(s(x1)), pos(s(x0))) we obtained the following new rules [LPAR04]:

EVAL(pos(s(s(y_1))), pos(s(x1))) → COND_EVAL(and(true, greatereq_int(pos(x1), pos(s(y_1)))), pos(s(s(y_1))), pos(s(x1)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                                        ↳ Instantiation
                                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                                              ↳ AND
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                  ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                                                                                      ↳ QReductionProof
                                                                                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                                                                                          ↳ ForwardInstantiation
                                                                                                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                                                                                                              ↳ QReductionProof

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

COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(y1)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL1(true, pos(s(z0)), pos(s(z1))) → EVAL(pos(s(z1)), pos(s(z1)))
EVAL(pos(s(s(y_1))), pos(s(x1))) → COND_EVAL(and(true, greatereq_int(pos(x1), pos(s(y_1)))), pos(s(s(y_1))), pos(s(x1)))

The TRS R consists of the following rules:

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

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)))

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

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



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ RemovalProof
                        ↳ RemovalProof
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Rewriting
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Rewriting
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Rewriting
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Rewriting
                                                          ↳ QDP
                                                            ↳ Rewriting
                                                              ↳ QDP
                                                                ↳ Rewriting
                                                                  ↳ QDP
                                                                    ↳ DependencyGraphProof
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Rewriting
                                                                                  ↳ QDP
                                                                                    ↳ Rewriting
                                                                                      ↳ QDP
                                                                                        ↳ Rewriting
                                                                                          ↳ QDP
                                                                                            ↳ Rewriting
                                                                                              ↳ QDP
                                                                                                ↳ Rewriting
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Rewriting
                                                                                                          ↳ QDP
                                                                                                            ↳ Rewriting
                                                                                                              ↳ QDP
                                                                                                                ↳ DependencyGraphProof
                                                                                                                  ↳ QDP
                                                                                                                    ↳ Rewriting
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Rewriting
                                                                                                                          ↳ QDP
                                                                                                                            ↳ Rewriting
                                                                                                                              ↳ QDP
                                                                                                                                ↳ Narrowing
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ Narrowing
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ Narrowing
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                  ↳ QDP
                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                      ↳ QDP
                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                          ↳ QDP
                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                    ↳ Narrowing
                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                            ↳ UsableRulesProof
                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                ↳ Narrowing
                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                    ↳ DependencyGraphProof
                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                            ↳ Instantiation
                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                    ↳ Rewriting
                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                        ↳ UsableRulesProof
                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                            ↳ Narrowing
                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                    ↳ UsableRulesProof
                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                        ↳ Narrowing
                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                              ↳ QDP
                                                                                                                                                                                                                                                                                ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                  ↳ QDP
                                                                                                                                                                                                                                                                                    ↳ QReductionProof
                                                                                                                                                                                                                                                                                      ↳ QDP
                                                                                                                                                                                                                                                                                        ↳ Instantiation
                                                                                                                                                                                                                                                                                          ↳ QDP
                                                                                                                                                                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                                                                                                                                                                              ↳ AND
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                ↳ QDP
                                                                                                                                                                                                                                                                                                  ↳ UsableRulesProof
                                                                                                                                                                                                                                                                                                    ↳ QDP
                                                                                                                                                                                                                                                                                                      ↳ QReductionProof
                                                                                                                                                                                                                                                                                                        ↳ QDP
                                                                                                                                                                                                                                                                                                          ↳ ForwardInstantiation
                                                                                                                                                                                                                                                                                                            ↳ QDP
                                                                                                                                                                                                                                                                                                              ↳ QReductionProof
                                                                                                                                                                                                                                                                                                                ↳ QDP

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

COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(y1)))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL1(and(true, greater_int(pos(x0), pos(x1))), pos(s(x0)), pos(s(x1)))
COND_EVAL1(true, pos(s(z0)), pos(s(z1))) → EVAL(pos(s(z1)), pos(s(z1)))
EVAL(pos(s(s(y_1))), pos(s(x1))) → COND_EVAL(and(true, greatereq_int(pos(x1), pos(s(y_1)))), pos(s(s(y_1))), pos(s(x1)))

The TRS R consists of the following rules:

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

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

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