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:

Cond_eval(TRUE, i, j, k) → eval(j, +@z(i, 1@z), -@z(k, 1@z))
eval(i, j, k) → Cond_eval(&&(<=@z(i, 100@z), <=@z(j, k)), i, j, k)

The set Q consists of the following terms:

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


Represented integers and predefined function symbols by Terms

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof

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

Cond_eval(true, i, j, k) → eval(j, plus_int(pos(s(0)), i), minus_int(k, pos(s(0))))
eval(i, j, k) → Cond_eval(and(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k)), i, j, k)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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:

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


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

COND_EVAL(true, i, j, k) → EVAL(j, plus_int(pos(s(0)), i), minus_int(k, pos(s(0))))
COND_EVAL(true, i, j, k) → PLUS_INT(pos(s(0)), i)
COND_EVAL(true, i, j, k) → MINUS_INT(k, pos(s(0)))
EVAL(i, j, k) → COND_EVAL(and(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k)), i, j, k)
EVAL(i, j, k) → AND(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k))
EVAL(i, j, k) → LESSEQ_INT(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
EVAL(i, j, k) → LESSEQ_INT(j, k)
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
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)
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 TRS R consists of the following rules:

Cond_eval(true, i, j, k) → eval(j, plus_int(pos(s(0)), i), minus_int(k, pos(s(0))))
eval(i, j, k) → Cond_eval(and(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k)), i, j, k)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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:

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

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

↳ ITRS
  ↳ ITRStoQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof

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

COND_EVAL(true, i, j, k) → EVAL(j, plus_int(pos(s(0)), i), minus_int(k, pos(s(0))))
COND_EVAL(true, i, j, k) → PLUS_INT(pos(s(0)), i)
COND_EVAL(true, i, j, k) → MINUS_INT(k, pos(s(0)))
EVAL(i, j, k) → COND_EVAL(and(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k)), i, j, k)
EVAL(i, j, k) → AND(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k))
EVAL(i, j, k) → LESSEQ_INT(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
EVAL(i, j, k) → LESSEQ_INT(j, k)
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
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)
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 TRS R consists of the following rules:

Cond_eval(true, i, j, k) → eval(j, plus_int(pos(s(0)), i), minus_int(k, pos(s(0))))
eval(i, j, k) → Cond_eval(and(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k)), i, j, k)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 13 less nodes.

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

Cond_eval(true, i, j, k) → eval(j, plus_int(pos(s(0)), i), minus_int(k, pos(s(0))))
eval(i, j, k) → Cond_eval(and(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k)), i, j, k)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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:

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

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

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:

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

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

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



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

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

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

Cond_eval(true, i, j, k) → eval(j, plus_int(pos(s(0)), i), minus_int(k, pos(s(0))))
eval(i, j, k) → Cond_eval(and(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k)), i, j, k)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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:

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

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

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:

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

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

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



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

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

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

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

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

The TRS R consists of the following rules:

Cond_eval(true, i, j, k) → eval(j, plus_int(pos(s(0)), i), minus_int(k, pos(s(0))))
eval(i, j, k) → Cond_eval(and(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k)), i, j, k)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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:

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

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

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

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

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

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

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

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



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

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

Cond_eval(true, i, j, k) → eval(j, plus_int(pos(s(0)), i), minus_int(k, pos(s(0))))
eval(i, j, k) → Cond_eval(and(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k)), i, j, k)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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:

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

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

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

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

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

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



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

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

EVAL(i, j, k) → COND_EVAL(and(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k)), i, j, k)
COND_EVAL(true, i, j, k) → EVAL(j, plus_int(pos(s(0)), i), minus_int(k, pos(s(0))))

The TRS R consists of the following rules:

Cond_eval(true, i, j, k) → eval(j, plus_int(pos(s(0)), i), minus_int(k, pos(s(0))))
eval(i, j, k) → Cond_eval(and(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k)), i, j, k)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
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))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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:

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

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

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

EVAL(i, j, k) → COND_EVAL(and(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k)), i, j, k)
COND_EVAL(true, i, j, k) → EVAL(j, plus_int(pos(s(0)), i), minus_int(k, pos(s(0))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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



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

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

EVAL(i, j, k) → COND_EVAL(and(lesseq_int(i, pos(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), lesseq_int(j, k)), i, j, k)
COND_EVAL(true, i, j, k) → EVAL(j, plus_int(pos(s(0)), i), minus_int(k, pos(s(0))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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