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_2(i, j) → Cond_eval_21(<=@z(j, -@z(i, 1@z)), i, j)
Cond_eval_2(TRUE, i, j) → eval_1(-@z(i, 1@z), j)
eval_2(i, j) → Cond_eval_2(>@z(j, -@z(i, 1@z)), i, j)
Cond_eval_21(TRUE, i, j) → eval_2(i, +@z(j, 1@z))
eval_1(i, j) → Cond_eval_1(>=@z(i, 0@z), i, j)
Cond_eval_1(TRUE, i, j) → eval_2(i, 0@z)

The set Q consists of the following terms:

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


Represented integers and predefined function symbols by Terms

↳ ITRS
  ↳ ITRStoQTRSProof
QTRS
      ↳ DependencyPairsProof

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

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

The set Q consists of the following terms:

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))


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

EVAL_2(i, j) → COND_EVAL_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
EVAL_2(i, j) → LESSEQ_INT(j, minus_int(i, pos(s(0))))
EVAL_2(i, j) → MINUS_INT(i, pos(s(0)))
COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
COND_EVAL_2(true, i, j) → MINUS_INT(i, pos(s(0)))
EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
EVAL_2(i, j) → GREATER_INT(j, minus_int(i, pos(s(0))))
COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
COND_EVAL_21(true, i, j) → PLUS_INT(pos(s(0)), j)
EVAL_1(i, j) → COND_EVAL_1(greatereq_int(i, pos(0)), i, j)
EVAL_1(i, j) → GREATEREQ_INT(i, pos(0))
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))
LESSEQ_INT(pos(s(x)), pos(s(y))) → LESSEQ_INT(pos(x), pos(y))
LESSEQ_INT(neg(s(x)), neg(s(y))) → LESSEQ_INT(neg(x), neg(y))
MINUS_INT(pos(x), pos(y)) → MINUS_NAT(x, y)
MINUS_INT(neg(x), neg(y)) → MINUS_NAT(y, x)
MINUS_INT(neg(x), pos(y)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
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)
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))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
QDP
          ↳ DependencyGraphProof

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

EVAL_2(i, j) → COND_EVAL_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
EVAL_2(i, j) → LESSEQ_INT(j, minus_int(i, pos(s(0))))
EVAL_2(i, j) → MINUS_INT(i, pos(s(0)))
COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
COND_EVAL_2(true, i, j) → MINUS_INT(i, pos(s(0)))
EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
EVAL_2(i, j) → GREATER_INT(j, minus_int(i, pos(s(0))))
COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
COND_EVAL_21(true, i, j) → PLUS_INT(pos(s(0)), j)
EVAL_1(i, j) → COND_EVAL_1(greatereq_int(i, pos(0)), i, j)
EVAL_1(i, j) → GREATEREQ_INT(i, pos(0))
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))
LESSEQ_INT(pos(s(x)), pos(s(y))) → LESSEQ_INT(pos(x), pos(y))
LESSEQ_INT(neg(s(x)), neg(s(y))) → LESSEQ_INT(neg(x), neg(y))
MINUS_INT(pos(x), pos(y)) → MINUS_NAT(x, y)
MINUS_INT(neg(x), neg(y)) → MINUS_NAT(y, x)
MINUS_INT(neg(x), pos(y)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
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)
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))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 9 SCCs with 14 less nodes.

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP
              ↳ 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_2(i, j) → Cond_eval_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
Cond_eval_2(true, i, j) → eval_1(minus_int(i, pos(s(0))), j)
eval_2(i, j) → Cond_eval_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
Cond_eval_21(true, i, j) → eval_2(i, plus_int(pos(s(0)), j))
eval_1(i, j) → Cond_eval_1(greatereq_int(i, pos(0)), i, j)
Cond_eval_1(true, i, j) → eval_2(i, pos(0))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))
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))

The set Q consists of the following terms:

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
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
              ↳ 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_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
QDP
                        ↳ UsableRulesReductionPairsProof
              ↳ QDP
              ↳ QDP
              ↳ 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
              ↳ 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
              ↳ 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_2(i, j) → Cond_eval_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
Cond_eval_2(true, i, j) → eval_1(minus_int(i, pos(s(0))), j)
eval_2(i, j) → Cond_eval_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
Cond_eval_21(true, i, j) → eval_2(i, plus_int(pos(s(0)), j))
eval_1(i, j) → Cond_eval_1(greatereq_int(i, pos(0)), i, j)
Cond_eval_1(true, i, j) → eval_2(i, pos(0))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))
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))

The set Q consists of the following terms:

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
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
              ↳ 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_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
QDP
                        ↳ UsableRulesReductionPairsProof
              ↳ QDP
              ↳ QDP
              ↳ 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
              ↳ 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
              ↳ 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_2(i, j) → Cond_eval_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
Cond_eval_2(true, i, j) → eval_1(minus_int(i, pos(s(0))), j)
eval_2(i, j) → Cond_eval_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
Cond_eval_21(true, i, j) → eval_2(i, plus_int(pos(s(0)), j))
eval_1(i, j) → Cond_eval_1(greatereq_int(i, pos(0)), i, j)
Cond_eval_1(true, i, j) → eval_2(i, pos(0))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))
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))

The set Q consists of the following terms:

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
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
              ↳ 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_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))



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

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

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

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

The following dependency pairs can be deleted:

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

Used ordering: POLO with Polynomial interpretation [POLO]:

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



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

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

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

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

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

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

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

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))



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

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

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

R is empty.
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
                ↳ 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
              ↳ 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_2(i, j) → Cond_eval_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
Cond_eval_2(true, i, j) → eval_1(minus_int(i, pos(s(0))), j)
eval_2(i, j) → Cond_eval_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
Cond_eval_21(true, i, j) → eval_2(i, plus_int(pos(s(0)), j))
eval_1(i, j) → Cond_eval_1(greatereq_int(i, pos(0)), i, j)
Cond_eval_1(true, i, j) → eval_2(i, pos(0))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))
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))

The set Q consists of the following terms:

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
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
              ↳ 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_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))



↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ 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
              ↳ 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_2(i, j) → Cond_eval_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
Cond_eval_2(true, i, j) → eval_1(minus_int(i, pos(s(0))), j)
eval_2(i, j) → Cond_eval_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
Cond_eval_21(true, i, j) → eval_2(i, plus_int(pos(s(0)), j))
eval_1(i, j) → Cond_eval_1(greatereq_int(i, pos(0)), i, j)
Cond_eval_1(true, i, j) → eval_2(i, pos(0))
lesseq_int(pos(0), pos(y)) → true
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))
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))

The set Q consists of the following terms:

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
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
              ↳ 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_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))



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

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

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

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

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

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

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))



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

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

LESSEQ_INT(neg(s(x)), neg(s(y))) → LESSEQ_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:

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

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(LESSEQ_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
              ↳ 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
              ↳ QDP
QDP
                ↳ UsableRulesProof
              ↳ QDP

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
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
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QReductionProof
              ↳ QDP

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

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

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

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))



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

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

LESSEQ_INT(pos(s(x)), pos(s(y))) → LESSEQ_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:

LESSEQ_INT(pos(s(x)), pos(s(y))) → LESSEQ_INT(pos(x), pos(y))
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(LESSEQ_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
              ↳ 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
              ↳ QDP
QDP
                ↳ UsableRulesProof

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

COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
EVAL_2(i, j) → COND_EVAL_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
EVAL_1(i, j) → COND_EVAL_1(greatereq_int(i, pos(0)), i, j)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
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
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QReductionProof

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

COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
EVAL_2(i, j) → COND_EVAL_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
EVAL_1(i, j) → COND_EVAL_1(greatereq_int(i, pos(0)), i, j)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

eval_2(x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_1(x0, x1)
Cond_eval_1(true, x0, x1)
lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
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))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

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



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

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

COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
EVAL_2(i, j) → COND_EVAL_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
EVAL_1(i, j) → COND_EVAL_1(greatereq_int(i, pos(0)), i, j)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL_1(neg(s(x0)), y1) → COND_EVAL_1(false, neg(s(x0)), y1)
EVAL_1(neg(0), y1) → COND_EVAL_1(true, neg(0), y1)
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)



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

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

COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
EVAL_2(i, j) → COND_EVAL_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))
EVAL_1(neg(s(x0)), y1) → COND_EVAL_1(false, neg(s(x0)), y1)
EVAL_1(neg(0), y1) → COND_EVAL_1(true, neg(0), y1)
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

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

EVAL_2(i, j) → COND_EVAL_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
EVAL_1(neg(0), y1) → COND_EVAL_1(true, neg(0), y1)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
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
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ UsableRulesProof
QDP
                                    ↳ QReductionProof

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

EVAL_2(i, j) → COND_EVAL_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
EVAL_1(neg(0), y1) → COND_EVAL_1(true, neg(0), y1)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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



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

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

EVAL_2(i, j) → COND_EVAL_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
EVAL_1(neg(0), y1) → COND_EVAL_1(true, neg(0), y1)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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_1(neg(0), y1) → COND_EVAL_1(true, neg(0), y1)
The remaining pairs can at least be oriented weakly.

EVAL_2(i, j) → COND_EVAL_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)
Used ordering: Matrix interpretation [MATRO]:

POL(EVAL_2(x1, x2)) =
/00\
\00/
·x1 +
/0\
\0/
+
/00\
\00/
·x2

POL(COND_EVAL_21(x1, x2, x3)) =
/00\
\00/
·x1 +
/0\
\0/
+
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(lesseq_int(x1, x2)) =
/00\
\00/
·x1 +
/0\
\0/
+
/00\
\00/
·x2

POL(minus_int(x1, x2)) =
/00\
\00/
·x1 +
/0\
\1/
+
/01\
\00/
·x2

POL(pos(x1)) =
/00\
\01/
·x1 +
/0\
\0/

POL(s(x1)) =
/00\
\00/
·x1 +
/0\
\0/

POL(0) =
/0\
\1/

POL(true) =
/1\
\1/

POL(plus_int(x1, x2)) =
/00\
\00/
·x1 +
/0\
\0/
+
/11\
\10/
·x2

POL(COND_EVAL_2(x1, x2, x3)) =
/00\
\00/
·x1 +
/0\
\0/
+
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(greater_int(x1, x2)) =
/00\
\00/
·x1 +
/0\
\0/
+
/00\
\00/
·x2

POL(EVAL_1(x1, x2)) =
/10\
\00/
·x1 +
/0\
\0/
+
/00\
\00/
·x2

POL(neg(x1)) =
/01\
\00/
·x1 +
/0\
\0/

POL(COND_EVAL_1(x1, x2, x3)) =
/00\
\00/
·x1 +
/0\
\0/
+
/00\
\00/
·x2 +
/00\
\00/
·x3

POL(false) =
/1\
\1/

POL(minus_nat(x1, x2)) =
/00\
\00/
·x1 +
/0\
\1/
+
/00\
\00/
·x2

POL(plus_nat(x1, x2)) =
/00\
\00/
·x1 +
/0\
\0/
+
/11\
\01/
·x2

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



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

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

EVAL_2(i, j) → COND_EVAL_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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: EVAL_2(i, j) → COND_EVAL_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
Positions in right side of the pair: Pair: COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
Positions in right side of the pair: Pair: EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
Positions in right side of the pair: Pair: COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
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
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ UsableRulesProof
                                  ↳ QDP
                                    ↳ QReductionProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ RemovalProof
QDP
                                            ↳ RemovalProof
                                            ↳ Narrowing

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

EVAL_2(i, j, x_removed) → COND_EVAL_21(lesseq_int(j, minus_int(i, x_removed)), i, j, x_removed)
COND_EVAL_21(true, i, j, x_removed) → EVAL_2(i, plus_int(x_removed, j), x_removed)
EVAL_2(i, j, x_removed) → COND_EVAL_2(greater_int(j, minus_int(i, x_removed)), i, j, x_removed)
COND_EVAL_2(true, i, j, x_removed) → EVAL_1(minus_int(i, x_removed), j, x_removed)
EVAL_1(pos(x0), y1, x_removed) → COND_EVAL_1(true, pos(x0), y1, x_removed)
COND_EVAL_1(true, i, j, x_removed) → EVAL_2(i, pos(0), x_removed)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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: EVAL_2(i, j) → COND_EVAL_21(lesseq_int(j, minus_int(i, pos(s(0)))), i, j)
Positions in right side of the pair: Pair: COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
Positions in right side of the pair: Pair: EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
Positions in right side of the pair: Pair: COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
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
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ UsableRulesProof
                                  ↳ QDP
                                    ↳ QReductionProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ RemovalProof
                                            ↳ RemovalProof
QDP
                                            ↳ Narrowing

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

EVAL_2(i, j, x_removed) → COND_EVAL_21(lesseq_int(j, minus_int(i, x_removed)), i, j, x_removed)
COND_EVAL_21(true, i, j, x_removed) → EVAL_2(i, plus_int(x_removed, j), x_removed)
EVAL_2(i, j, x_removed) → COND_EVAL_2(greater_int(j, minus_int(i, x_removed)), i, j, x_removed)
COND_EVAL_2(true, i, j, x_removed) → EVAL_1(minus_int(i, x_removed), j, x_removed)
EVAL_1(pos(x0), y1, x_removed) → COND_EVAL_1(true, pos(x0), y1, x_removed)
COND_EVAL_1(true, i, j, x_removed) → EVAL_2(i, pos(0), x_removed)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL_2(neg(x0), y1) → COND_EVAL_21(lesseq_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)
EVAL_2(pos(x0), y1) → COND_EVAL_21(lesseq_int(y1, minus_nat(x0, s(0))), pos(x0), y1)



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

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

COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
EVAL_2(i, j) → COND_EVAL_2(greater_int(j, minus_int(i, pos(s(0)))), i, j)
COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)
EVAL_2(neg(x0), y1) → COND_EVAL_21(lesseq_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)
EVAL_2(pos(x0), y1) → COND_EVAL_21(lesseq_int(y1, minus_nat(x0, s(0))), pos(x0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL_2(pos(x0), y1) → COND_EVAL_2(greater_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
EVAL_2(neg(x0), y1) → COND_EVAL_2(greater_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)



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

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

COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)
EVAL_2(neg(x0), y1) → COND_EVAL_21(lesseq_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)
EVAL_2(pos(x0), y1) → COND_EVAL_21(lesseq_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
EVAL_2(pos(x0), y1) → COND_EVAL_2(greater_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
EVAL_2(neg(x0), y1) → COND_EVAL_2(greater_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL_2(true, i, j) → EVAL_1(minus_int(i, pos(s(0))), j) at position [0] we obtained the following new rules [LPAR04]:

COND_EVAL_2(true, neg(x0), y1) → EVAL_1(neg(plus_nat(x0, s(0))), y1)
COND_EVAL_2(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)



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

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

COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)
EVAL_2(neg(x0), y1) → COND_EVAL_21(lesseq_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)
EVAL_2(pos(x0), y1) → COND_EVAL_21(lesseq_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
EVAL_2(pos(x0), y1) → COND_EVAL_2(greater_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
EVAL_2(neg(x0), y1) → COND_EVAL_2(greater_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)
COND_EVAL_2(true, neg(x0), y1) → EVAL_1(neg(plus_nat(x0, s(0))), y1)
COND_EVAL_2(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

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

EVAL_2(neg(x0), y1) → COND_EVAL_21(lesseq_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)
COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
EVAL_2(pos(x0), y1) → COND_EVAL_21(lesseq_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
EVAL_2(pos(x0), y1) → COND_EVAL_2(greater_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
COND_EVAL_2(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ UsableRulesProof
                                  ↳ QDP
                                    ↳ QReductionProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ RemovalProof
                                            ↳ RemovalProof
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ UsableRulesProof
QDP
                                                                ↳ QReductionProof

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

EVAL_2(neg(x0), y1) → COND_EVAL_21(lesseq_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)
COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
EVAL_2(pos(x0), y1) → COND_EVAL_21(lesseq_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
EVAL_2(pos(x0), y1) → COND_EVAL_2(greater_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
COND_EVAL_2(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
lesseq_int(pos(0), pos(y)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
plus_int(pos(x), neg(y)) → minus_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))

The set Q consists of the following terms:

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

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

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



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

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

EVAL_2(neg(x0), y1) → COND_EVAL_21(lesseq_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)
COND_EVAL_21(true, i, j) → EVAL_2(i, plus_int(pos(s(0)), j))
EVAL_2(pos(x0), y1) → COND_EVAL_21(lesseq_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
EVAL_2(pos(x0), y1) → COND_EVAL_2(greater_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
COND_EVAL_2(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
lesseq_int(pos(0), pos(y)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
plus_int(pos(x), neg(y)) → minus_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))

The set Q consists of the following terms:

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

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

COND_EVAL_21(true, neg(z0), z1) → EVAL_2(neg(z0), plus_int(pos(s(0)), z1))
COND_EVAL_21(true, pos(z0), z1) → EVAL_2(pos(z0), plus_int(pos(s(0)), z1))



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

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

EVAL_2(neg(x0), y1) → COND_EVAL_21(lesseq_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)
EVAL_2(pos(x0), y1) → COND_EVAL_21(lesseq_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
EVAL_2(pos(x0), y1) → COND_EVAL_2(greater_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
COND_EVAL_2(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))
COND_EVAL_21(true, neg(z0), z1) → EVAL_2(neg(z0), plus_int(pos(s(0)), z1))
COND_EVAL_21(true, pos(z0), z1) → EVAL_2(pos(z0), plus_int(pos(s(0)), z1))

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
lesseq_int(pos(0), pos(y)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
plus_int(pos(x), neg(y)) → minus_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))

The set Q consists of the following terms:

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

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

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

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

COND_EVAL_21(true, neg(z0), z1) → EVAL_2(neg(z0), plus_int(pos(s(0)), z1))
EVAL_2(neg(x0), y1) → COND_EVAL_21(lesseq_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
lesseq_int(pos(0), pos(y)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
plus_int(pos(x), neg(y)) → minus_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))

The set Q consists of the following terms:

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

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
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ UsableRulesProof
                                  ↳ QDP
                                    ↳ QReductionProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ RemovalProof
                                            ↳ RemovalProof
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ UsableRulesProof
                                                              ↳ QDP
                                                                ↳ QReductionProof
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ DependencyGraphProof
                                                                          ↳ AND
                                                                            ↳ QDP
                                                                              ↳ UsableRulesProof
QDP
                                                                                  ↳ QReductionProof
                                                                            ↳ QDP

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

COND_EVAL_21(true, neg(z0), z1) → EVAL_2(neg(z0), plus_int(pos(s(0)), z1))
EVAL_2(neg(x0), y1) → COND_EVAL_21(lesseq_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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



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

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

COND_EVAL_21(true, neg(z0), z1) → EVAL_2(neg(z0), plus_int(pos(s(0)), z1))
EVAL_2(neg(x0), y1) → COND_EVAL_21(lesseq_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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_21(true, neg(z0), z1) → EVAL_2(neg(z0), plus_int(pos(s(0)), z1))
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
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ UsableRulesProof
                                  ↳ QDP
                                    ↳ QReductionProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ RemovalProof
                                            ↳ RemovalProof
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ UsableRulesProof
                                                              ↳ QDP
                                                                ↳ QReductionProof
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ DependencyGraphProof
                                                                          ↳ AND
                                                                            ↳ QDP
                                                                              ↳ UsableRulesProof
                                                                                ↳ QDP
                                                                                  ↳ QReductionProof
                                                                                    ↳ QDP
                                                                                      ↳ RemovalProof
QDP
                                                                                      ↳ RemovalProof
                                                                            ↳ QDP

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

COND_EVAL_21(true, neg(z0), z1, x_removed) → EVAL_2(neg(z0), plus_int(x_removed, z1), x_removed)
EVAL_2(neg(x0), y1, x_removed) → COND_EVAL_21(lesseq_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1, x_removed)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

lesseq_int(pos(0), pos(x0))
lesseq_int(pos(0), neg(0))
lesseq_int(neg(x0), pos(x1))
lesseq_int(neg(x0), neg(0))
lesseq_int(pos(x0), neg(s(x1)))
lesseq_int(neg(0), neg(s(x0)))
lesseq_int(pos(s(x0)), pos(0))
lesseq_int(pos(s(x0)), neg(x1))
lesseq_int(pos(s(x0)), pos(s(x1)))
lesseq_int(neg(s(x0)), neg(s(x1)))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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_21(true, neg(z0), z1) → EVAL_2(neg(z0), plus_int(pos(s(0)), z1))
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
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
                  ↳ QDP
                    ↳ QReductionProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ UsableRulesProof
                                  ↳ QDP
                                    ↳ QReductionProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ RemovalProof
                                            ↳ RemovalProof
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ UsableRulesProof
                                                              ↳ QDP
                                                                ↳ QReductionProof
                                                                  ↳ QDP
                                                                    ↳ Instantiation
                                                                      ↳ QDP
                                                                        ↳ DependencyGraphProof
                                                                          ↳ AND
                                                                            ↳ QDP
                                                                              ↳ UsableRulesProof
                                                                                ↳ QDP
                                                                                  ↳ QReductionProof
                                                                                    ↳ QDP
                                                                                      ↳ RemovalProof
                                                                                      ↳ RemovalProof
QDP
                                                                            ↳ QDP

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

COND_EVAL_21(true, neg(z0), z1, x_removed) → EVAL_2(neg(z0), plus_int(x_removed, z1), x_removed)
EVAL_2(neg(x0), y1, x_removed) → COND_EVAL_21(lesseq_int(y1, neg(plus_nat(x0, s(0)))), neg(x0), y1, x_removed)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

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

EVAL_2(pos(x0), y1) → COND_EVAL_21(lesseq_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
COND_EVAL_21(true, pos(z0), z1) → EVAL_2(pos(z0), plus_int(pos(s(0)), z1))
EVAL_2(pos(x0), y1) → COND_EVAL_2(greater_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
COND_EVAL_2(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)
COND_EVAL_1(true, i, j) → EVAL_2(i, pos(0))

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
lesseq_int(pos(0), pos(y)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
plus_int(pos(x), neg(y)) → minus_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))

The set Q consists of the following terms:

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

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

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



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

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

EVAL_2(pos(x0), y1) → COND_EVAL_21(lesseq_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
COND_EVAL_21(true, pos(z0), z1) → EVAL_2(pos(z0), plus_int(pos(s(0)), z1))
EVAL_2(pos(x0), y1) → COND_EVAL_2(greater_int(y1, minus_nat(x0, s(0))), pos(x0), y1)
COND_EVAL_2(true, pos(x0), y1) → EVAL_1(minus_nat(x0, s(0)), y1)
EVAL_1(pos(x0), y1) → COND_EVAL_1(true, pos(x0), y1)
COND_EVAL_1(true, pos(z0), z1) → EVAL_2(pos(z0), pos(0))

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
lesseq_int(pos(0), neg(0)) → true
lesseq_int(neg(x), neg(0)) → true
lesseq_int(pos(x), neg(s(y))) → false
lesseq_int(neg(0), neg(s(y))) → false
lesseq_int(pos(s(x)), neg(y)) → false
lesseq_int(neg(s(x)), neg(s(y))) → lesseq_int(neg(x), neg(y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
lesseq_int(pos(0), pos(y)) → true
lesseq_int(neg(x), pos(y)) → true
lesseq_int(pos(s(x)), pos(0)) → false
lesseq_int(pos(s(x)), pos(s(y))) → lesseq_int(pos(x), pos(y))
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
plus_int(pos(x), neg(y)) → minus_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))

The set Q consists of the following terms:

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

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