Termination of the given ITRSProblem could not be shown:



ITRS
  ↳ ITRStoQTRSProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(&&(&&(>@z(+@z(x, y), z), >=@z(z, 0@z)), >@z(x, 0@z)), x, y, z)
Cond_eval2(TRUE, x, y, z) → eval(x, y, z)
eval(x, y, z) → Cond_eval1(&&(&&(&&(>@z(+@z(x, y), z), >=@z(z, 0@z)), >=@z(0@z, x)), >@z(y, 0@z)), x, y, z)
Cond_eval(TRUE, x, y, z) → eval(-@z(x, 1@z), y, z)
Cond_eval1(TRUE, x, y, z) → eval(x, -@z(y, 1@z), z)
eval(x, y, z) → Cond_eval2(&&(&&(&&(>@z(+@z(x, y), z), >=@z(z, 0@z)), >=@z(0@z, x)), >=@z(0@z, y)), x, y, z)

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval2(TRUE, x0, x1, x2)
Cond_eval(TRUE, x0, x1, x2)
Cond_eval1(TRUE, x0, x1, x2)


Represented integers and predefined function symbols by Terms

↳ ITRS
  ↳ ITRStoQTRSProof
QTRS
      ↳ DependencyPairsProof

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

eval(x, y, z) → Cond_eval(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
Cond_eval2(true, x, y, z) → eval(x, y, z)
eval(x, y, z) → Cond_eval1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(minus_int(x, pos(s(0))), y, z)
Cond_eval1(true, x, y, z) → eval(x, minus_int(y, pos(s(0))), z)
eval(x, y, z) → Cond_eval2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))


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

EVAL(x, y, z) → COND_EVAL(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
EVAL(x, y, z) → AND(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0)))
EVAL(x, y, z) → AND(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0)))
EVAL(x, y, z) → GREATER_INT(plus_int(x, y), z)
EVAL(x, y, z) → PLUS_INT(x, y)
EVAL(x, y, z) → GREATEREQ_INT(z, pos(0))
EVAL(x, y, z) → GREATER_INT(x, pos(0))
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(x, y, z) → COND_EVAL1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
EVAL(x, y, z) → AND(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0)))
EVAL(x, y, z) → AND(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x))
EVAL(x, y, z) → GREATEREQ_INT(pos(0), x)
EVAL(x, y, z) → GREATER_INT(y, pos(0))
COND_EVAL(true, x, y, z) → EVAL(minus_int(x, pos(s(0))), y, z)
COND_EVAL(true, x, y, z) → MINUS_INT(x, pos(s(0)))
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL1(true, x, y, z) → MINUS_INT(y, pos(s(0)))
EVAL(x, y, z) → COND_EVAL2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
EVAL(x, y, z) → AND(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y))
EVAL(x, y, z) → GREATEREQ_INT(pos(0), y)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
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)

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
Cond_eval2(true, x, y, z) → eval(x, y, z)
eval(x, y, z) → Cond_eval1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(minus_int(x, pos(s(0))), y, z)
Cond_eval1(true, x, y, z) → eval(x, minus_int(y, pos(s(0))), z)
eval(x, y, z) → Cond_eval2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
QDP
          ↳ DependencyGraphProof

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

EVAL(x, y, z) → COND_EVAL(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
EVAL(x, y, z) → AND(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0)))
EVAL(x, y, z) → AND(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0)))
EVAL(x, y, z) → GREATER_INT(plus_int(x, y), z)
EVAL(x, y, z) → PLUS_INT(x, y)
EVAL(x, y, z) → GREATEREQ_INT(z, pos(0))
EVAL(x, y, z) → GREATER_INT(x, pos(0))
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(x, y, z) → COND_EVAL1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
EVAL(x, y, z) → AND(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0)))
EVAL(x, y, z) → AND(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x))
EVAL(x, y, z) → GREATEREQ_INT(pos(0), x)
EVAL(x, y, z) → GREATER_INT(y, pos(0))
COND_EVAL(true, x, y, z) → EVAL(minus_int(x, pos(s(0))), y, z)
COND_EVAL(true, x, y, z) → MINUS_INT(x, pos(s(0)))
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL1(true, x, y, z) → MINUS_INT(y, pos(s(0)))
EVAL(x, y, z) → COND_EVAL2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
EVAL(x, y, z) → AND(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y))
EVAL(x, y, z) → GREATEREQ_INT(pos(0), y)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
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)

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
Cond_eval2(true, x, y, z) → eval(x, y, z)
eval(x, y, z) → Cond_eval1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(minus_int(x, pos(s(0))), y, z)
Cond_eval1(true, x, y, z) → eval(x, minus_int(y, pos(s(0))), z)
eval(x, y, z) → Cond_eval2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 22 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:

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

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
Cond_eval2(true, x, y, z) → eval(x, y, z)
eval(x, y, z) → Cond_eval1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(minus_int(x, pos(s(0))), y, z)
Cond_eval1(true, x, y, z) → eval(x, minus_int(y, pos(s(0))), z)
eval(x, y, z) → Cond_eval2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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:

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

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

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))



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

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

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
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ UsableRulesReductionPairsProof
QDP
                            ↳ PisEmptyProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

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

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

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

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
Cond_eval2(true, x, y, z) → eval(x, y, z)
eval(x, y, z) → Cond_eval1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(minus_int(x, pos(s(0))), y, z)
Cond_eval1(true, x, y, z) → eval(x, minus_int(y, pos(s(0))), z)
eval(x, y, z) → Cond_eval2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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:

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

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

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))



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

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

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
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ UsableRulesReductionPairsProof
QDP
                            ↳ PisEmptyProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

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

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

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

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
Cond_eval2(true, x, y, z) → eval(x, y, z)
eval(x, y, z) → Cond_eval1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(minus_int(x, pos(s(0))), y, z)
Cond_eval1(true, x, y, z) → eval(x, minus_int(y, pos(s(0))), z)
eval(x, y, z) → Cond_eval2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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:

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

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

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))



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

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
Cond_eval2(true, x, y, z) → eval(x, y, z)
eval(x, y, z) → Cond_eval1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(minus_int(x, pos(s(0))), y, z)
Cond_eval1(true, x, y, z) → eval(x, minus_int(y, pos(s(0))), z)
eval(x, y, z) → Cond_eval2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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:

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
QDP
                        ↳ QDPSizeChangeProof
              ↳ 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
              ↳ 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, z) → Cond_eval(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
Cond_eval2(true, x, y, z) → eval(x, y, z)
eval(x, y, z) → Cond_eval1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(minus_int(x, pos(s(0))), y, z)
Cond_eval1(true, x, y, z) → eval(x, minus_int(y, pos(s(0))), z)
eval(x, y, z) → Cond_eval2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))



↳ 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, z) → Cond_eval(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
Cond_eval2(true, x, y, z) → eval(x, y, z)
eval(x, y, z) → Cond_eval1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(minus_int(x, pos(s(0))), y, z)
Cond_eval1(true, x, y, z) → eval(x, minus_int(y, pos(s(0))), z)
eval(x, y, z) → Cond_eval2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))



↳ 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, z) → EVAL(minus_int(x, pos(s(0))), y, z)
EVAL(x, y, z) → COND_EVAL(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
EVAL(x, y, z) → COND_EVAL1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
EVAL(x, y, z) → COND_EVAL2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
Cond_eval2(true, x, y, z) → eval(x, y, z)
eval(x, y, z) → Cond_eval1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
Cond_eval(true, x, y, z) → eval(minus_int(x, pos(s(0))), y, z)
Cond_eval1(true, x, y, z) → eval(x, minus_int(y, pos(s(0))), z)
eval(x, y, z) → Cond_eval2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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, z) → EVAL(minus_int(x, pos(s(0))), y, z)
EVAL(x, y, z) → COND_EVAL(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
EVAL(x, y, z) → COND_EVAL1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
EVAL(x, y, z) → COND_EVAL2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(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))

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

eval(x0, x1, x2)
Cond_eval2(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
Cond_eval1(true, x0, x1, x2)



↳ 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, z) → EVAL(minus_int(x, pos(s(0))), y, z)
EVAL(x, y, z) → COND_EVAL(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z)
EVAL(x, y, z) → COND_EVAL1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
EVAL(x, y, z) → COND_EVAL2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)

The TRS R consists of the following rules:

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

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

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, z) → EVAL(minus_int(x, pos(s(0))), y, z)
Positions in right side of the pair: Pair: COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
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, z, x_removed) → EVAL(minus_int(x, x_removed), y, z, x_removed)
EVAL(x, y, z, x_removed) → COND_EVAL(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z, x_removed)
EVAL(x, y, z, x_removed) → COND_EVAL1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z, x_removed)
EVAL(x, y, z, x_removed) → COND_EVAL2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z, x_removed)
COND_EVAL1(true, x, y, z, x_removed) → EVAL(x, minus_int(y, x_removed), z, x_removed)
COND_EVAL2(true, x, y, z, x_removed) → EVAL(x, y, z, x_removed)

The TRS R consists of the following rules:

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

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

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, z) → EVAL(minus_int(x, pos(s(0))), y, z)
Positions in right side of the pair: Pair: COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
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, z, x_removed) → EVAL(minus_int(x, x_removed), y, z, x_removed)
EVAL(x, y, z, x_removed) → COND_EVAL(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greater_int(x, pos(0))), x, y, z, x_removed)
EVAL(x, y, z, x_removed) → COND_EVAL1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z, x_removed)
EVAL(x, y, z, x_removed) → COND_EVAL2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z, x_removed)
COND_EVAL1(true, x, y, z, x_removed) → EVAL(x, minus_int(y, x_removed), z, x_removed)
COND_EVAL2(true, x, y, z, x_removed) → EVAL(x, y, z, x_removed)

The TRS R consists of the following rules:

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

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

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

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



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

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

COND_EVAL(true, x, y, z) → EVAL(minus_int(x, pos(s(0))), y, z)
EVAL(x, y, z) → COND_EVAL1(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, z)
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
EVAL(x, y, z) → COND_EVAL2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)

The TRS R consists of the following rules:

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

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

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

EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)



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

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

COND_EVAL(true, x, y, z) → EVAL(minus_int(x, pos(s(0))), y, z)
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
EVAL(x, y, z) → COND_EVAL2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2) at position [0,0,1] we obtained the following new rules [LPAR04]:

EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)



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

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

COND_EVAL(true, x, y, z) → EVAL(minus_int(x, pos(s(0))), y, z)
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
EVAL(x, y, z) → COND_EVAL2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2) at position [0,0,1] we obtained the following new rules [LPAR04]:

EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)



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

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

COND_EVAL(true, x, y, z) → EVAL(minus_int(x, pos(s(0))), y, z)
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
EVAL(x, y, z) → COND_EVAL2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(x, y, z) → COND_EVAL2(and(and(and(greater_int(plus_int(x, y), z), greatereq_int(z, pos(0))), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, z) at position [0] we obtained the following new rules [LPAR04]:

EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1), y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), neg(x1))), pos(x0), neg(x1), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)



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

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

COND_EVAL(true, x, y, z) → EVAL(minus_int(x, pos(s(0))), y, z)
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1), y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), neg(x1))), pos(x0), neg(x1), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1), y2) at position [0,0,1] we obtained the following new rules [LPAR04]:

EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1), y2)



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

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

COND_EVAL(true, x, y, z) → EVAL(minus_int(x, pos(s(0))), y, z)
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), neg(x1))), pos(x0), neg(x1), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1), y2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2) at position [0,0,1] we obtained the following new rules [LPAR04]:

EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)



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

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

COND_EVAL(true, x, y, z) → EVAL(minus_int(x, pos(s(0))), y, z)
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), neg(x1))), pos(x0), neg(x1), y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), neg(x1))), pos(x0), neg(x1), y2) at position [0,1] we obtained the following new rules [LPAR04]:

EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)



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

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

COND_EVAL(true, x, y, z) → EVAL(minus_int(x, pos(s(0))), y, z)
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1), y2) at position [0,1] we obtained the following new rules [LPAR04]:

EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)



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

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

COND_EVAL(true, x, y, z) → EVAL(minus_int(x, pos(s(0))), y, z)
COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)

The TRS R consists of the following rules:

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

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

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

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



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

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

COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_int(pos(z0), pos(s(0))), neg(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_int(pos(s(z0)), pos(s(0))), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(minus_int(neg(0), pos(s(0))), z0, z1)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(minus_int(neg(z0), pos(s(0))), neg(z1), z2)
COND_EVAL(true, pos(0), z0, z1) → EVAL(minus_int(pos(0), pos(s(0))), z0, z1)
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(minus_int(neg(z0), pos(s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1), z2)
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1, z2)

The TRS R consists of the following rules:

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

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

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

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



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

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

COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_int(pos(s(z0)), pos(s(0))), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(minus_int(neg(0), pos(s(0))), z0, z1)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(minus_int(neg(z0), pos(s(0))), neg(z1), z2)
COND_EVAL(true, pos(0), z0, z1) → EVAL(minus_int(pos(0), pos(s(0))), z0, z1)
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(minus_int(neg(z0), pos(s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1), z2)
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1, z2)
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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

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



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

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

COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(minus_int(neg(0), pos(s(0))), z0, z1)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(minus_int(neg(z0), pos(s(0))), neg(z1), z2)
COND_EVAL(true, pos(0), z0, z1) → EVAL(minus_int(pos(0), pos(s(0))), z0, z1)
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(minus_int(neg(z0), pos(s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1), z2)
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1, z2)
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(s(z0), s(0)), z1, z2)

The TRS R consists of the following rules:

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

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

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

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



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

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

COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(minus_int(neg(z0), pos(s(0))), neg(z1), z2)
COND_EVAL(true, pos(0), z0, z1) → EVAL(minus_int(pos(0), pos(s(0))), z0, z1)
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(minus_int(neg(z0), pos(s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1), z2)
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1, z2)
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(s(z0), s(0)), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(plus_nat(0, s(0))), z0, z1)

The TRS R consists of the following rules:

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

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

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

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



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

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

COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, pos(0), z0, z1) → EVAL(minus_int(pos(0), pos(s(0))), z0, z1)
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(minus_int(neg(z0), pos(s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1), z2)
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1, z2)
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(s(z0), s(0)), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(plus_nat(0, s(0))), z0, z1)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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

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



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

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

COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(minus_int(neg(z0), pos(s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1), z2)
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1, z2)
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(s(z0), s(0)), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(plus_nat(0, s(0))), z0, z1)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, pos(0), z0, z1) → EVAL(minus_nat(0, s(0)), z0, z1)

The TRS R consists of the following rules:

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

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

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

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



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

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

COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1), z2)
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1, z2)
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(s(z0), s(0)), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(plus_nat(0, s(0))), z0, z1)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, pos(0), z0, z1) → EVAL(minus_nat(0, s(0)), z0, z1)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)

The TRS R consists of the following rules:

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

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

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

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



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

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

COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1, z2)
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(s(z0), s(0)), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(plus_nat(0, s(0))), z0, z1)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, pos(0), z0, z1) → EVAL(minus_nat(0, s(0)), z0, z1)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)

The TRS R consists of the following rules:

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

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

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

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



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

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

COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(s(z0), s(0)), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(plus_nat(0, s(0))), z0, z1)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, pos(0), z0, z1) → EVAL(minus_nat(0, s(0)), z0, z1)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(plus_nat(s(z0), s(0))), z1, z2)

The TRS R consists of the following rules:

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

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

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

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



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

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

COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(plus_nat(0, s(0))), z0, z1)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, pos(0), z0, z1) → EVAL(minus_nat(0, s(0)), z0, z1)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(plus_nat(s(z0), s(0))), z1, z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)

The TRS R consists of the following rules:

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

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

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

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



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

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

COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, pos(0), z0, z1) → EVAL(minus_nat(0, s(0)), z0, z1)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(plus_nat(s(z0), s(0))), z1, z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)

The TRS R consists of the following rules:

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

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

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

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



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

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

COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(plus_nat(s(z0), s(0))), z1, z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(plus_nat(s(z0), s(0))), z1, z2) at position [0,0] we obtained the following new rules [LPAR04]:

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



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

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

COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z)
COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL1(true, x, y, z) → EVAL(x, minus_int(y, pos(s(0))), z) we obtained the following new rules [LPAR04]:

COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), minus_int(neg(z1), pos(s(0))), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_int(pos(s(z1)), pos(s(0))), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_int(pos(0), pos(s(0))), z1)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0))), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), minus_int(neg(z1), pos(s(0))), z2)
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, minus_int(neg(s(z1)), pos(s(0))), z2)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, minus_int(neg(0), pos(s(0))), z1)



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

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

COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), minus_int(neg(z1), pos(s(0))), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_int(pos(s(z1)), pos(s(0))), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_int(pos(0), pos(s(0))), z1)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0))), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), minus_int(neg(z1), pos(s(0))), z2)
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, minus_int(neg(s(z1)), pos(s(0))), z2)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, minus_int(neg(0), pos(s(0))), z1)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), minus_int(neg(z1), pos(s(0))), z2) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)



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

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

COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_int(pos(s(z1)), pos(s(0))), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_int(pos(0), pos(s(0))), z1)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0))), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), minus_int(neg(z1), pos(s(0))), z2)
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, minus_int(neg(s(z1)), pos(s(0))), z2)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, minus_int(neg(0), pos(s(0))), z1)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_int(pos(s(z1)), pos(s(0))), z2) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(s(z1), s(0)), z2)



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

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

COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_int(pos(0), pos(s(0))), z1)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0))), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), minus_int(neg(z1), pos(s(0))), z2)
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, minus_int(neg(s(z1)), pos(s(0))), z2)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, minus_int(neg(0), pos(s(0))), z1)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(s(z1), s(0)), z2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_int(pos(0), pos(s(0))), z1) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_nat(0, s(0)), z1)



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

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

COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0))), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), minus_int(neg(z1), pos(s(0))), z2)
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, minus_int(neg(s(z1)), pos(s(0))), z2)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, minus_int(neg(0), pos(s(0))), z1)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(s(z1), s(0)), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_nat(0, s(0)), z1)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))), z2) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)



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

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

COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0))), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), minus_int(neg(z1), pos(s(0))), z2)
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, minus_int(neg(s(z1)), pos(s(0))), z2)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, minus_int(neg(0), pos(s(0))), z1)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(s(z1), s(0)), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_nat(0, s(0)), z1)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0))), z2) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)



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

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

COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), minus_int(neg(z1), pos(s(0))), z2)
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, minus_int(neg(s(z1)), pos(s(0))), z2)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, minus_int(neg(0), pos(s(0))), z1)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(s(z1), s(0)), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_nat(0, s(0)), z1)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), minus_int(neg(z1), pos(s(0))), z2) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)



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

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

COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, minus_int(neg(s(z1)), pos(s(0))), z2)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, minus_int(neg(0), pos(s(0))), z1)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(s(z1), s(0)), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_nat(0, s(0)), z1)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, minus_int(neg(s(z1)), pos(s(0))), z2) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(plus_nat(s(z1), s(0))), z2)



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

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

COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, minus_int(neg(0), pos(s(0))), z1)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(s(z1), s(0)), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_nat(0, s(0)), z1)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(plus_nat(s(z1), s(0))), z2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, minus_int(neg(0), pos(s(0))), z1) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(plus_nat(0, s(0))), z1)



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

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

COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(s(z1), s(0)), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_nat(0, s(0)), z1)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(plus_nat(s(z1), s(0))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(plus_nat(0, s(0))), z1)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(s(z1), s(0)), z2) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(z1, 0), z2)



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

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

COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_nat(0, s(0)), z1)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(plus_nat(s(z1), s(0))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(plus_nat(0, s(0))), z1)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(z1, 0), z2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, minus_nat(0, s(0)), z1) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, neg(s(0)), z1)



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

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

COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(plus_nat(s(z1), s(0))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(plus_nat(0, s(0))), z1)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(z1, 0), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, neg(s(0)), z1)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(plus_nat(s(z1), s(0))), z2) at position [1,0] we obtained the following new rules [LPAR04]:

COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)



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

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

COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(plus_nat(0, s(0))), z1)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(z1, 0), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(plus_nat(0, s(0))), z1) at position [1,0] we obtained the following new rules [LPAR04]:

COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)



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

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

COND_EVAL2(true, x, y, z) → EVAL(x, y, z)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(z1, 0), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)

The TRS R consists of the following rules:

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

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL2(true, x, y, z) → EVAL(x, y, z) we obtained the following new rules [LPAR04]:

COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)



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

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

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(z1, 0), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.


COND_EVAL(true, pos(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
The remaining pairs can at least be oriented weakly.

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(z1, 0), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(COND_EVAL(x1, x2, x3, x4)) = x2   
POL(COND_EVAL1(x1, x2, x3, x4)) = x2   
POL(COND_EVAL2(x1, x2, x3, x4)) = x2   
POL(EVAL(x1, x2, x3)) = x1   
POL(and(x1, x2)) = 0   
POL(false) = 0   
POL(greater_int(x1, x2)) = 0   
POL(greatereq_int(x1, x2)) = 0   
POL(minus_int(x1, x2)) = x1   
POL(minus_nat(x1, x2)) = 1   
POL(neg(x1)) = 0   
POL(plus_int(x1, x2)) = 0   
POL(plus_nat(x1, x2)) = 0   
POL(pos(x1)) = 1   
POL(s(x1)) = 0   
POL(true) = 0   

The following usable rules [FROCOS05] were oriented:

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



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

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

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(z1, 0), z2)
COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.


COND_EVAL1(true, z0, pos(0), z1) → EVAL(z0, neg(s(0)), z1)
The remaining pairs can at least be oriented weakly.

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(z1, 0), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1   
POL(COND_EVAL(x1, x2, x3, x4)) = x3   
POL(COND_EVAL1(x1, x2, x3, x4)) = x3   
POL(COND_EVAL2(x1, x2, x3, x4)) = x3   
POL(EVAL(x1, x2, x3)) = x2   
POL(and(x1, x2)) = 0   
POL(false) = 0   
POL(greater_int(x1, x2)) = 0   
POL(greatereq_int(x1, x2)) = 1 + x2   
POL(minus_int(x1, x2)) = x1   
POL(minus_nat(x1, x2)) = x1   
POL(neg(x1)) = 0   
POL(plus_int(x1, x2)) = 1 + x1 + x2   
POL(plus_nat(x1, x2)) = 0   
POL(pos(x1)) = x1   
POL(s(x1)) = x1   
POL(true) = 0   

The following usable rules [FROCOS05] were oriented:

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



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

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

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(z1, 0), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.


COND_EVAL1(true, z0, pos(s(z1)), z2) → EVAL(z0, minus_nat(z1, 0), z2)
The remaining pairs can at least be oriented weakly.

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(COND_EVAL(x1, x2, x3, x4)) = x3   
POL(COND_EVAL1(x1, x2, x3, x4)) = x3   
POL(COND_EVAL2(x1, x2, x3, x4)) = x3   
POL(EVAL(x1, x2, x3)) = x2   
POL(and(x1, x2)) = 0   
POL(false) = 0   
POL(greater_int(x1, x2)) = 0   
POL(greatereq_int(x1, x2)) = 0   
POL(minus_int(x1, x2)) = x1   
POL(minus_nat(x1, x2)) = x1   
POL(neg(x1)) = 0   
POL(plus_int(x1, x2)) = 0   
POL(plus_nat(x1, x2)) = 0   
POL(pos(x1)) = x1   
POL(s(x1)) = 1 + x1   
POL(true) = 0   

The following usable rules [FROCOS05] were oriented:

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



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

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

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.


COND_EVAL(true, pos(s(z0)), z1, z2) → EVAL(minus_nat(z0, 0), z1, z2)
The remaining pairs can at least be oriented weakly.

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(COND_EVAL(x1, x2, x3, x4)) = x2   
POL(COND_EVAL1(x1, x2, x3, x4)) = x2   
POL(COND_EVAL2(x1, x2, x3, x4)) = x2   
POL(EVAL(x1, x2, x3)) = x1   
POL(and(x1, x2)) = 0   
POL(false) = 0   
POL(greater_int(x1, x2)) = 0   
POL(greatereq_int(x1, x2)) = 0   
POL(minus_int(x1, x2)) = x1   
POL(minus_nat(x1, x2)) = x1   
POL(neg(x1)) = 0   
POL(plus_int(x1, x2)) = 0   
POL(plus_nat(x1, x2)) = 0   
POL(pos(x1)) = x1   
POL(s(x1)) = 1 + x1   
POL(true) = 0   

The following usable rules [FROCOS05] were oriented:

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



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

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

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.


COND_EVAL(true, neg(z0), neg(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), neg(z1), z2)
COND_EVAL(true, neg(z0), pos(z1), z2) → EVAL(neg(plus_nat(z0, s(0))), pos(z1), z2)
COND_EVAL(true, neg(0), z0, z1) → EVAL(neg(s(0)), z0, z1)
COND_EVAL(true, neg(s(z0)), z1, z2) → EVAL(neg(s(plus_nat(z0, s(0)))), z1, z2)
The remaining pairs can at least be oriented weakly.

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(COND_EVAL(x1, x2, x3, x4)) = x1   
POL(COND_EVAL1(x1, x2, x3, x4)) = x2   
POL(COND_EVAL2(x1, x2, x3, x4)) = x2   
POL(EVAL(x1, x2, x3)) = x1   
POL(and(x1, x2)) = x2   
POL(false) = 0   
POL(greater_int(x1, x2)) = x1   
POL(greatereq_int(x1, x2)) = 0   
POL(minus_int(x1, x2)) = 1   
POL(minus_nat(x1, x2)) = 1   
POL(neg(x1)) = 0   
POL(plus_int(x1, x2)) = 0   
POL(plus_nat(x1, x2)) = 0   
POL(pos(x1)) = x1   
POL(s(x1)) = 1   
POL(true) = 1   

The following usable rules [FROCOS05] were oriented:

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



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

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

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.


EVAL(neg(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(0), y1), y2), greatereq_int(y2, pos(0))), false), neg(0), y1, y2)
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(neg(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), neg(s(x0)), y1, y2)
The remaining pairs can at least be oriented weakly.

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(COND_EVAL(x1, x2, x3, x4)) = x1   
POL(COND_EVAL1(x1, x2, x3, x4)) = x2   
POL(COND_EVAL2(x1, x2, x3, x4)) = x2   
POL(EVAL(x1, x2, x3)) = x1   
POL(and(x1, x2)) = x2   
POL(false) = 0   
POL(greater_int(x1, x2)) = x1   
POL(greatereq_int(x1, x2)) = 0   
POL(minus_int(x1, x2)) = 1   
POL(minus_nat(x1, x2)) = 1   
POL(neg(x1)) = 1   
POL(plus_int(x1, x2)) = 0   
POL(plus_nat(x1, x2)) = 0   
POL(pos(x1)) = x1   
POL(s(x1)) = 1   
POL(true) = 1   

The following usable rules [FROCOS05] were oriented:

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



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

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

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.


COND_EVAL1(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(plus_nat(z1, s(0))), z2)
COND_EVAL1(true, z0, neg(s(z1)), z2) → EVAL(z0, neg(s(plus_nat(z1, s(0)))), z2)
COND_EVAL1(true, z0, neg(0), z1) → EVAL(z0, neg(s(0)), z1)
The remaining pairs can at least be oriented weakly.

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(COND_EVAL(x1, x2, x3, x4)) = x3   
POL(COND_EVAL1(x1, x2, x3, x4)) = x1   
POL(COND_EVAL2(x1, x2, x3, x4)) = x3   
POL(EVAL(x1, x2, x3)) = x2   
POL(and(x1, x2)) = x2   
POL(false) = 0   
POL(greater_int(x1, x2)) = x1   
POL(greatereq_int(x1, x2)) = 0   
POL(minus_int(x1, x2)) = 1   
POL(minus_nat(x1, x2)) = 1   
POL(neg(x1)) = 0   
POL(plus_int(x1, x2)) = 0   
POL(plus_nat(x1, x2)) = 0   
POL(pos(x1)) = x1   
POL(s(x1)) = 1   
POL(true) = 1   

The following usable rules [FROCOS05] were oriented:

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



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

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

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.


EVAL(pos(0), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), false), pos(0), y1, y2)
The remaining pairs can at least be oriented weakly.

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(COND_EVAL(x1, x2, x3, x4)) = x1   
POL(COND_EVAL1(x1, x2, x3, x4)) = x2   
POL(COND_EVAL2(x1, x2, x3, x4)) = x2   
POL(EVAL(x1, x2, x3)) = x1   
POL(and(x1, x2)) = x2   
POL(false) = 0   
POL(greater_int(x1, x2)) = x1   
POL(greatereq_int(x1, x2)) = 0   
POL(minus_int(x1, x2)) = x2   
POL(minus_nat(x1, x2)) = 1   
POL(neg(x1)) = 0   
POL(plus_int(x1, x2)) = 0   
POL(plus_nat(x1, x2)) = 0   
POL(pos(x1)) = 1   
POL(s(x1)) = 0   
POL(true) = 1   

The following usable rules [FROCOS05] were oriented:

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



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

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

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.


EVAL(y0, pos(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(0), y2)
The remaining pairs can at least be oriented weakly.

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1   
POL(COND_EVAL(x1, x2, x3, x4)) = x3   
POL(COND_EVAL1(x1, x2, x3, x4)) = x1   
POL(COND_EVAL2(x1, x2, x3, x4)) = x3   
POL(EVAL(x1, x2, x3)) = x2   
POL(and(x1, x2)) = x2   
POL(false) = 0   
POL(greater_int(x1, x2)) = x1   
POL(greatereq_int(x1, x2)) = 0   
POL(minus_int(x1, x2)) = 1   
POL(minus_nat(x1, x2)) = 1   
POL(neg(x1)) = 0   
POL(plus_int(x1, x2)) = 0   
POL(plus_nat(x1, x2)) = 0   
POL(pos(x1)) = x1   
POL(s(x1)) = 1   
POL(true) = 1   

The following usable rules [FROCOS05] were oriented:

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



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

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

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.


EVAL(y0, pos(s(x0)), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)), y2)
The remaining pairs can at least be oriented weakly.

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(COND_EVAL(x1, x2, x3, x4)) = x1 + x2 + x4   
POL(COND_EVAL1(x1, x2, x3, x4)) = 1 + x2 + x4   
POL(COND_EVAL2(x1, x2, x3, x4)) = x1 + x2 + x4   
POL(EVAL(x1, x2, x3)) = 1 + x1 + x3   
POL(and(x1, x2)) = x2   
POL(false) = 0   
POL(greater_int(x1, x2)) = x2   
POL(greatereq_int(x1, x2)) = x1   
POL(minus_int(x1, x2)) = x1   
POL(minus_nat(x1, x2)) = 1   
POL(neg(x1)) = 1   
POL(plus_int(x1, x2)) = 0   
POL(plus_nat(x1, x2)) = 0   
POL(pos(x1)) = 1   
POL(s(x1)) = 0   
POL(true) = 1   

The following usable rules [FROCOS05] were oriented:

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



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

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

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.


EVAL(y0, neg(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)), y2)
EVAL(y0, neg(0), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, neg(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), false), y0, neg(0), y2)
The remaining pairs can at least be oriented weakly.

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(COND_EVAL(x1, x2, x3, x4)) = x3   
POL(COND_EVAL1(x1, x2, x3, x4)) = x1   
POL(COND_EVAL2(x1, x2, x3, x4)) = x3   
POL(EVAL(x1, x2, x3)) = x2   
POL(and(x1, x2)) = x2   
POL(false) = 0   
POL(greater_int(x1, x2)) = x1   
POL(greatereq_int(x1, x2)) = 0   
POL(minus_int(x1, x2)) = 1   
POL(minus_nat(x1, x2)) = 1   
POL(neg(x1)) = 1   
POL(plus_int(x1, x2)) = 0   
POL(plus_nat(x1, x2)) = 0   
POL(pos(x1)) = x1   
POL(s(x1)) = 1   
POL(true) = 1   

The following usable rules [FROCOS05] were oriented:

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



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

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

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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


The following pairs can be oriented strictly and are deleted.


EVAL(neg(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1), y2)
The remaining pairs can at least be oriented weakly.

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(EVAL(x1, x2, x3)) =
/0A0A\
\0A0A/
·x1 +
/1A\
\1A/
+
/0A0A\
\0A0A/
·x2 +
/-I-I\
\-I-I/
·x3

POL(neg(x1)) =
/-I-I\
\-I-I/
·x1 +
/0A\
\0A/

POL(pos(x1)) =
/0A-I\
\-I0A/
·x1 +
/-I\
\-I/

POL(COND_EVAL(x1, x2, x3, x4)) =
/0A0A\
\-I0A/
·x1 +
/1A\
\-I/
+
/0A0A\
\0A0A/
·x2 +
/0A0A\
\0A0A/
·x3 +
/-I-I\
\-I-I/
·x4

POL(and(x1, x2)) =
/0A-I\
\-I-I/
·x1 +
/0A\
\-I/
+
/0A-I\
\-I0A/
·x2

POL(greater_int(x1, x2)) =
/-I0A\
\0A0A/
·x1 +
/0A\
\-I/
+
/-I-I\
\-I1A/
·x2

POL(minus_nat(x1, x2)) =
/0A0A\
\-I0A/
·x1 +
/0A\
\0A/
+
/-I-I\
\-I-I/
·x2

POL(greatereq_int(x1, x2)) =
/-I-I\
\0A0A/
·x1 +
/0A\
\0A/
+
/-I0A\
\1A0A/
·x2

POL(0) =
/0A\
\-I/

POL(plus_nat(x1, x2)) =
/-I0A\
\0A0A/
·x1 +
/1A\
\0A/
+
/0A0A\
\0A0A/
·x2

POL(plus_int(x1, x2)) =
/0A0A\
\0A0A/
·x1 +
/1A\
\0A/
+
/0A0A\
\0A0A/
·x2

POL(true) =
/0A\
\1A/

POL(s(x1)) =
/0A0A\
\0A0A/
·x1 +
/-I\
\1A/

POL(false) =
/0A\
\0A/

POL(COND_EVAL1(x1, x2, x3, x4)) =
/0A0A\
\0A0A/
·x1 +
/0A\
\0A/
+
/0A0A\
\0A0A/
·x2 +
/0A0A\
\0A0A/
·x3 +
/-I-I\
\-I-I/
·x4

POL(COND_EVAL2(x1, x2, x3, x4)) =
/0A-I\
\0A-I/
·x1 +
/1A\
\1A/
+
/0A0A\
\0A0A/
·x2 +
/0A0A\
\0A0A/
·x3 +
/-I-I\
\-I-I/
·x4

POL(minus_int(x1, x2)) =
/0A0A\
\0A0A/
·x1 +
/0A\
\0A/
+
/0A-I\
\0A-I/
·x2

The following usable rules [FROCOS05] were oriented:

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



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

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

EVAL(neg(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), pos(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(neg(x0), pos(0))), neg(x0), neg(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, pos(x0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greater_int(y0, pos(0))), y0, y1, pos(x0))
EVAL(y0, y1, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greater_int(y0, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greater_int(y0, pos(0))), y0, y1, neg(s(x0)))
EVAL(pos(x0), neg(x1), y2) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greater_int(pos(x0), pos(0))), pos(x0), neg(x1), y2)
EVAL(pos(s(x0)), y1, y2) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), true), pos(s(x0)), y1, y2)
EVAL(neg(x1), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), neg(x1), y1, y2)
EVAL(y0, pos(s(x0)), y2) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, pos(s(x0))), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)), y2)
EVAL(pos(0), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greater_int(y1, pos(0))), pos(0), y1, y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1), y2)
EVAL(y0, y1, neg(s(x0))) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(s(x0)))
EVAL(y0, y1, neg(0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greater_int(y1, pos(0))), y0, y1, pos(x0))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL1(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greater_int(y1, pos(0))), pos(s(x0)), y1, y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1), y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL1(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1), y2)
EVAL(pos(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(pos(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1), y2)
EVAL(neg(x1), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(neg(x1), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), neg(x1), y1, y2)
EVAL(y0, pos(0), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, pos(0)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, pos(0), y2)
EVAL(pos(0), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(0), y1), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), y1)), pos(0), y1, y2)
EVAL(y0, neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, neg(x1)), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), y0)), true), y0, neg(x1), y2)
EVAL(y0, y1, neg(0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(0))
EVAL(y0, y1, pos(x0)) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), pos(x0)), true), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, pos(x0))
EVAL(y0, y1, neg(s(x0))) → COND_EVAL2(and(and(and(greater_int(plus_int(y0, y1), neg(s(x0))), false), greatereq_int(pos(0), y0)), greatereq_int(pos(0), y1)), y0, y1, neg(s(x0)))
EVAL(pos(s(x0)), y1, y2) → COND_EVAL2(and(and(and(greater_int(plus_int(pos(s(x0)), y1), y2), greatereq_int(y2, pos(0))), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1, y2)
EVAL(neg(x0), pos(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x1, x0), y2), greatereq_int(y2, pos(0))), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1), y2)
EVAL(pos(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(minus_nat(x0, x1), y2), greatereq_int(y2, pos(0))), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1), y2)
EVAL(neg(x0), neg(x1), y2) → COND_EVAL2(and(and(and(greater_int(neg(plus_nat(x0, x1)), y2), greatereq_int(y2, pos(0))), true), true), neg(x0), neg(x1), y2)
COND_EVAL(true, z0, z1, pos(z2)) → EVAL(minus_int(z0, pos(s(0))), z1, pos(z2))
COND_EVAL(true, z0, z1, neg(0)) → EVAL(minus_int(z0, pos(s(0))), z1, neg(0))
COND_EVAL(true, z0, z1, neg(s(z2))) → EVAL(minus_int(z0, pos(s(0))), z1, neg(s(z2)))
COND_EVAL(true, pos(z0), neg(z1), z2) → EVAL(minus_nat(z0, s(0)), neg(z1), z2)
COND_EVAL(true, pos(z0), pos(z1), z2) → EVAL(minus_nat(z0, s(0)), pos(z1), z2)
COND_EVAL1(true, neg(z0), z1, z2) → EVAL(neg(z0), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, z0, z1, neg(s(z2))) → EVAL(z0, minus_int(z1, pos(s(0))), neg(s(z2)))
COND_EVAL1(true, z0, z1, neg(0)) → EVAL(z0, minus_int(z1, pos(s(0))), neg(0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), pos(z2))
COND_EVAL1(true, pos(0), z0, z1) → EVAL(pos(0), minus_int(z0, pos(s(0))), z1)
COND_EVAL1(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))), z2)
COND_EVAL1(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), minus_nat(z1, s(0)), z2)
COND_EVAL1(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), minus_nat(z1, s(0)), z2)
COND_EVAL2(true, neg(z0), z1, z2) → EVAL(neg(z0), z1, z2)
COND_EVAL2(true, z0, neg(z1), z2) → EVAL(z0, neg(z1), z2)
COND_EVAL2(true, pos(z0), pos(z1), z2) → EVAL(pos(z0), pos(z1), z2)
COND_EVAL2(true, z0, pos(0), z1) → EVAL(z0, pos(0), z1)
COND_EVAL2(true, pos(0), z0, z1) → EVAL(pos(0), z0, z1)
COND_EVAL2(true, z0, pos(s(z1)), z2) → EVAL(z0, pos(s(z1)), z2)
COND_EVAL2(true, z0, z1, neg(0)) → EVAL(z0, z1, neg(0))
COND_EVAL2(true, z0, z1, pos(z2)) → EVAL(z0, z1, pos(z2))
COND_EVAL2(true, z0, z1, neg(s(z2))) → EVAL(z0, z1, neg(s(z2)))
COND_EVAL2(true, pos(s(z0)), z1, z2) → EVAL(pos(s(z0)), z1, z2)
COND_EVAL2(true, neg(z0), pos(z1), z2) → EVAL(neg(z0), pos(z1), z2)
COND_EVAL2(true, pos(z0), neg(z1), z2) → EVAL(pos(z0), neg(z1), z2)
COND_EVAL2(true, neg(z0), neg(z1), z2) → EVAL(neg(z0), neg(z1), z2)

The TRS R consists of the following rules:

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

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

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