Termination proof of /tmp/tmpYY

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, x) → eval(+@z(*@z(-@z(2@z), x), 10@z))
eval(x) → Cond_eval(>=@z(x, 0@z), x)

The set Q consists of the following terms:

Cond_eval(TRUE, x0)
eval(x0)

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, x) → eval(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))
eval(x) → Cond_eval(greatereq_int(x, pos(0)), x)
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)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
neg_int(pos(x)) → neg(x)
neg_int(neg(x)) → pos(x)
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:

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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:

COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))
COND_EVAL(true, x) → PLUS_INT(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x))
COND_EVAL(true, x) → MULT_INT(neg_int(pos(s(s(0)))), x)
COND_EVAL(true, x) → NEG_INT(pos(s(s(0))))
EVAL(x) → COND_EVAL(greatereq_int(x, pos(0)), x)
EVAL(x) → GREATEREQ_INT(x, pos(0))
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)
MULT_INT(pos(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(pos(x), neg(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), neg(y)) → MULT_NAT(x, y)
MULT_NAT(s(x), s(y)) → PLUS_NAT(mult_nat(x, s(y)), s(y))
MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(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:

Cond_eval(true, x) → eval(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))
eval(x) → Cond_eval(greatereq_int(x, pos(0)), x)
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)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
neg_int(pos(x)) → neg(x)
neg_int(neg(x)) → pos(x)
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:

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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:

COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))
COND_EVAL(true, x) → PLUS_INT(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x))
COND_EVAL(true, x) → MULT_INT(neg_int(pos(s(s(0)))), x)
COND_EVAL(true, x) → NEG_INT(pos(s(s(0))))
EVAL(x) → COND_EVAL(greatereq_int(x, pos(0)), x)
EVAL(x) → GREATEREQ_INT(x, pos(0))
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)
MULT_INT(pos(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(pos(x), neg(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), neg(y)) → MULT_NAT(x, y)
MULT_NAT(s(x), s(y)) → PLUS_NAT(mult_nat(x, s(y)), s(y))
MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(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:

Cond_eval(true, x) → eval(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))
eval(x) → Cond_eval(greatereq_int(x, pos(0)), x)
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)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
neg_int(pos(x)) → neg(x)
neg_int(neg(x)) → pos(x)
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:

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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 6 SCCs with 13 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ 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:

Cond_eval(true, x) → eval(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))
eval(x) → Cond_eval(greatereq_int(x, pos(0)), x)
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)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
neg_int(pos(x)) → neg(x)
neg_int(neg(x)) → pos(x)
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:

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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

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

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

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

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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].

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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

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

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

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

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

The TRS R consists of the following rules:

Cond_eval(true, x) → eval(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))
eval(x) → Cond_eval(greatereq_int(x, pos(0)), x)
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)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
neg_int(pos(x)) → neg(x)
neg_int(neg(x)) → pos(x)
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:

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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].

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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

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

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

Cond_eval(true, x) → eval(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))
eval(x) → Cond_eval(greatereq_int(x, pos(0)), x)
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)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
neg_int(pos(x)) → neg(x)
neg_int(neg(x)) → pos(x)
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:

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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

              ↳ 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)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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].

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

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

MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
The graph contains the following edges 1 > 1, 2 > 2

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

Cond_eval(true, x) → eval(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))
eval(x) → Cond_eval(greatereq_int(x, pos(0)), x)
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)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
neg_int(pos(x)) → neg(x)
neg_int(neg(x)) → pos(x)
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:

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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

              ↳ 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)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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].

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
The graph contains the following edges 1 > 1, 2 >= 2

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))

The TRS R consists of the following rules:

Cond_eval(true, x) → eval(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))
eval(x) → Cond_eval(greatereq_int(x, pos(0)), x)
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)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
neg_int(pos(x)) → neg(x)
neg_int(neg(x)) → pos(x)
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:

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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

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

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))

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

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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].

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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

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

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))

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

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))
The graph contains the following edges 1 > 1, 2 >= 2

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                ↳ UsableRulesProof

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

EVAL(x) → COND_EVAL(greatereq_int(x, pos(0)), x)
COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))

The TRS R consists of the following rules:

Cond_eval(true, x) → eval(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))
eval(x) → Cond_eval(greatereq_int(x, pos(0)), x)
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)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
neg_int(pos(x)) → neg(x)
neg_int(neg(x)) → pos(x)
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:

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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

                ↳ UsableRulesProof

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

EVAL(x) → COND_EVAL(greatereq_int(x, pos(0)), x)
COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
neg_int(pos(x)) → neg(x)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

The set Q consists of the following terms:

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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].

Cond_eval(true, x0)
eval(x0)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                ↳ UsableRulesProof

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

EVAL(x) → COND_EVAL(greatereq_int(x, pos(0)), x)
COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
neg_int(pos(x)) → neg(x)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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 rewriting [LPAR04] the rule COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x))) at position [0,1,0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg(s(s(0))), x)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                ↳ UsableRulesProof

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

EVAL(x) → COND_EVAL(greatereq_int(x, pos(0)), x)
COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg(s(s(0))), x)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
neg_int(pos(x)) → neg(x)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                ↳ UsableRulesProof

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

EVAL(x) → COND_EVAL(greatereq_int(x, pos(0)), x)
COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg(s(s(0))), x)))

The TRS R consists of the following rules:

mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false

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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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].

neg_int(pos(x0))
neg_int(neg(x0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                ↳ UsableRulesProof

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

EVAL(x) → COND_EVAL(greatereq_int(x, pos(0)), x)
COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg(s(s(0))), x)))

The TRS R consists of the following rules:

mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg(s(s(0))), x)))
EVAL(neg(0)) → COND_EVAL(true, neg(0))
EVAL(pos(x0)) → COND_EVAL(true, pos(x0))
EVAL(neg(s(x0))) → COND_EVAL(false, neg(s(x0)))

The TRS R consists of the following rules:

mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false

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

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                ↳ UsableRulesProof

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

EVAL(neg(0)) → COND_EVAL(true, neg(0))
COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg(s(s(0))), x)))
EVAL(pos(x0)) → COND_EVAL(true, pos(x0))

The TRS R consists of the following rules:

mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                ↳ UsableRulesProof

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

EVAL(neg(0)) → COND_EVAL(true, neg(0))
COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg(s(s(0))), x)))
EVAL(pos(x0)) → COND_EVAL(true, pos(x0))

The TRS R consists of the following rules:

mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

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

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                ↳ UsableRulesProof

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

EVAL(neg(0)) → COND_EVAL(true, neg(0))
COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg(s(s(0))), x)))
EVAL(pos(x0)) → COND_EVAL(true, pos(x0))

The TRS R consists of the following rules:

mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

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

The following pairs can be oriented strictly and are deleted.

EVAL(neg(0)) → COND_EVAL(true, neg(0))

The remaining pairs can at least be oriented weakly.

COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg(s(s(0))), x)))
EVAL(pos(x0)) → COND_EVAL(true, pos(x0))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1
POL(COND_EVAL(x₁, x₂)) = 0
POL(EVAL(x₁)) = x₁
POL(minus_nat(x₁, x₂)) = 0
POL(mult_int(x₁, x₂)) = 0
POL(mult_nat(x₁, x₂)) = 0
POL(neg(x₁)) = x₁
POL(plus_int(x₁, x₂)) = 0
POL(plus_nat(x₁, x₂)) = 1 + x₂
POL(pos(x₁)) = 0
POL(s(x₁)) = 0
POL(true) = 0

The following usable rules [FROCOS05] were oriented:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                ↳ UsableRulesProof

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

COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg(s(s(0))), x)))
EVAL(pos(x0)) → COND_EVAL(true, pos(x0))

The TRS R consists of the following rules:

mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

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

COND_EVAL(true, pos(x1)) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), neg(mult_nat(s(s(0)), x1))))
COND_EVAL(true, neg(x1)) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), pos(mult_nat(s(s(0)), x1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

EVAL(pos(x0)) → COND_EVAL(true, pos(x0))
COND_EVAL(true, pos(x1)) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), neg(mult_nat(s(s(0)), x1))))
COND_EVAL(true, neg(x1)) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), pos(mult_nat(s(s(0)), x1))))

The TRS R consists of the following rules:

mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                ↳ UsableRulesProof

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

COND_EVAL(true, pos(x1)) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), neg(mult_nat(s(s(0)), x1))))
EVAL(pos(x0)) → COND_EVAL(true, pos(x0))

The TRS R consists of the following rules:

mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                ↳ UsableRulesProof

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

COND_EVAL(true, pos(x1)) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), neg(mult_nat(s(s(0)), x1))))
EVAL(pos(x0)) → COND_EVAL(true, pos(x0))

The TRS R consists of the following rules:

mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

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

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                ↳ UsableRulesProof

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

COND_EVAL(true, pos(x1)) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), neg(mult_nat(s(s(0)), x1))))
EVAL(pos(x0)) → COND_EVAL(true, pos(x0))

The TRS R consists of the following rules:

mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

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))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

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

COND_EVAL(true, pos(x1)) → EVAL(minus_nat(s(s(s(s(s(s(s(s(s(s(0)))))))))), mult_nat(s(s(0)), x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                ↳ UsableRulesProof

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

EVAL(pos(x0)) → COND_EVAL(true, pos(x0))
COND_EVAL(true, pos(x1)) → EVAL(minus_nat(s(s(s(s(s(s(s(s(s(s(0)))))))))), mult_nat(s(s(0)), x1)))

The TRS R consists of the following rules:

mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))

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))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                ↳ UsableRulesProof

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

EVAL(pos(x0)) → COND_EVAL(true, pos(x0))
COND_EVAL(true, pos(x1)) → EVAL(minus_nat(s(s(s(s(s(s(s(s(s(s(0)))))))))), mult_nat(s(s(0)), x1)))

The TRS R consists of the following rules:

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

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))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

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

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ MNOCProof

                ↳ UsableRulesProof

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

EVAL(pos(x0)) → COND_EVAL(true, pos(x0))
COND_EVAL(true, pos(x1)) → EVAL(minus_nat(s(s(s(s(s(s(s(s(s(s(0)))))))))), mult_nat(s(s(0)), x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ MNOCProof

                                                                                      ↳ QDP

                ↳ UsableRulesProof

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

EVAL(pos(x0)) → COND_EVAL(true, pos(x0))
COND_EVAL(true, pos(x1)) → EVAL(minus_nat(s(s(s(s(s(s(s(s(s(s(0)))))))))), mult_nat(s(s(0)), x1)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all (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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

EVAL(x) → COND_EVAL(greatereq_int(x, pos(0)), x)
COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
neg_int(pos(x)) → neg(x)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

The set Q consists of the following terms:

Cond_eval(true, x0)
eval(x0)
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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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].

Cond_eval(true, x0)
eval(x0)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

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

EVAL(x) → COND_EVAL(greatereq_int(x, pos(0)), x)
COND_EVAL(true, x) → EVAL(plus_int(pos(s(s(s(s(s(s(s(s(s(s(0))))))))))), mult_int(neg_int(pos(s(s(0)))), x)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
neg_int(pos(x)) → neg(x)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
plus_int(pos(x), neg(y)) → minus_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)
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

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))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
neg_int(pos(x0))
neg_int(neg(x0))
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.