Termination proof of /tmp/tmpmSdIDq/a.05.itrs

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, y) → eval(x, +@z(y, 1@z))
eval(x, y) → Cond_eval(>=@z(x, +@z(y, 1@z)), x, y)

The set Q consists of the following terms:

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

Represented integers and predefined function symbols by Terms

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

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

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

The set Q consists of the following terms:

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

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, y) → EVAL(x, plus_int(pos(s(0)), y))
COND_EVAL(true, x, y) → PLUS_INT(pos(s(0)), y)
EVAL(x, y) → COND_EVAL(greatereq_int(x, plus_int(pos(s(0)), y)), x, y)
EVAL(x, y) → GREATEREQ_INT(x, plus_int(pos(s(0)), y))
EVAL(x, y) → PLUS_INT(pos(s(0)), y)
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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, y) → EVAL(x, plus_int(pos(s(0)), y))
COND_EVAL(true, x, y) → PLUS_INT(pos(s(0)), y)
EVAL(x, y) → COND_EVAL(greatereq_int(x, plus_int(pos(s(0)), y)), x, y)
EVAL(x, y) → GREATEREQ_INT(x, plus_int(pos(s(0)), y))
EVAL(x, y) → PLUS_INT(pos(s(0)), y)
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 7 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

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

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

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

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

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

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

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

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

                ↳ UsableRulesProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                ↳ UsableRulesProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                ↳ UsableRulesProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: EVAL(x, y) → COND_EVAL(greatereq_int(x, plus_int(pos(s(0)), y)), x, y)
Positions in right side of the pair:

[0,1,0]

Pair: COND_EVAL(true, x, y) → EVAL(x, plus_int(pos(s(0)), y))
Positions in right side of the pair:

[1,0]

The new variable was added to all pairs as a new argument[CONREM].

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                        ↳ RemovalProof

                        ↳ Narrowing

                ↳ UsableRulesProof

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

EVAL(x, y, x_removed) → COND_EVAL(greatereq_int(x, plus_int(x_removed, y)), x, y, x_removed)
COND_EVAL(true, x, y, x_removed) → EVAL(x, plus_int(x_removed, y), x_removed)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

[0,1,0]

Pair: COND_EVAL(true, x, y) → EVAL(x, plus_int(pos(s(0)), y))
Positions in right side of the pair:

[1,0]

The new variable was added to all pairs as a new argument[CONREM].

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                          ↳ QDP

                        ↳ Narrowing

                ↳ UsableRulesProof

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

EVAL(x, y, x_removed) → COND_EVAL(greatereq_int(x, plus_int(x_removed, y)), x, y, x_removed)
COND_EVAL(true, x, y, x_removed) → EVAL(x, plus_int(x_removed, y), x_removed)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(plus_nat(s(0), x1))), y0, pos(x1))
EVAL(y0, neg(x1)) → COND_EVAL(greatereq_int(y0, minus_nat(s(0), x1)), y0, neg(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                ↳ UsableRulesProof

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

COND_EVAL(true, x, y) → EVAL(x, plus_int(pos(s(0)), y))
EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(plus_nat(s(0), x1))), y0, pos(x1))
EVAL(y0, neg(x1)) → COND_EVAL(greatereq_int(y0, minus_nat(s(0), x1)), y0, neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(plus_nat(s(0), x1))), y0, pos(x1)) at position [0,1,0] we obtained the following new rules [LPAR04]:

EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(plus_nat(0, x1)))), y0, pos(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                ↳ UsableRulesProof

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

COND_EVAL(true, x, y) → EVAL(x, plus_int(pos(s(0)), y))
EVAL(y0, neg(x1)) → COND_EVAL(greatereq_int(y0, minus_nat(s(0), x1)), y0, neg(x1))
EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(plus_nat(0, x1)))), y0, pos(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(plus_nat(0, x1)))), y0, pos(x1)) at position [0,1,0,0] we obtained the following new rules [LPAR04]:

EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(x1))), y0, pos(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                ↳ UsableRulesProof

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

COND_EVAL(true, x, y) → EVAL(x, plus_int(pos(s(0)), y))
EVAL(y0, neg(x1)) → COND_EVAL(greatereq_int(y0, minus_nat(s(0), x1)), y0, neg(x1))
EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(x1))), y0, pos(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL(true, x, y) → EVAL(x, plus_int(pos(s(0)), y)) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(plus_nat(s(0), x1)))
COND_EVAL(true, y0, neg(x1)) → EVAL(y0, minus_nat(s(0), x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

EVAL(y0, neg(x1)) → COND_EVAL(greatereq_int(y0, minus_nat(s(0), x1)), y0, neg(x1))
EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(x1))), y0, pos(x1))
COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(plus_nat(s(0), x1)))
COND_EVAL(true, y0, neg(x1)) → EVAL(y0, minus_nat(s(0), x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                            ↳ QDP

                ↳ UsableRulesProof

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

COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(plus_nat(s(0), x1)))
EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(x1))), y0, pos(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                            ↳ QDP

                ↳ UsableRulesProof

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

COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(plus_nat(s(0), x1)))
EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(x1))), y0, pos(x1))

The TRS R consists of the following rules:

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

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))
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].

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Rewriting

                                            ↳ QDP

                ↳ UsableRulesProof

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

COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(plus_nat(s(0), x1)))
EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(x1))), y0, pos(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), 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 rewriting [LPAR04] the rule COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(plus_nat(s(0), x1))) at position [1,0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(s(plus_nat(0, x1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                            ↳ QDP

                ↳ UsableRulesProof

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

EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(x1))), y0, pos(x1))
COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(s(plus_nat(0, x1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                            ↳ QDP

                ↳ UsableRulesProof

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

EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(x1))), y0, pos(x1))
COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(s(plus_nat(0, x1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), 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 rewriting [LPAR04] the rule COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(s(plus_nat(0, x1)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                            ↳ QDP

                ↳ UsableRulesProof

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

EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(x1))), y0, pos(x1))
COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                            ↳ QDP

                ↳ UsableRulesProof

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

EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(x1))), y0, pos(x1))
COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

plus_nat(0, x0)
plus_nat(s(x0), 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].

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Instantiation

                                            ↳ QDP

                ↳ UsableRulesProof

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

EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(x1))), y0, pos(x1))
COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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 instantiating [LPAR04] the rule EVAL(y0, pos(x1)) → COND_EVAL(greatereq_int(y0, pos(s(x1))), y0, pos(x1)) we obtained the following new rules [LPAR04]:

EVAL(z0, pos(s(z1))) → COND_EVAL(greatereq_int(z0, pos(s(s(z1)))), z0, pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Instantiation

                                                                            ↳ QDP

                                                                              ↳ Instantiation

                                            ↳ QDP

                ↳ UsableRulesProof

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

COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(s(x1)))
EVAL(z0, pos(s(z1))) → COND_EVAL(greatereq_int(z0, pos(s(s(z1)))), z0, pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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 instantiating [LPAR04] the rule COND_EVAL(true, y0, pos(x1)) → EVAL(y0, pos(s(x1))) we obtained the following new rules [LPAR04]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Instantiation

                                                                            ↳ QDP

                                                                              ↳ Instantiation

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                            ↳ QDP

                ↳ UsableRulesProof

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

EVAL(z0, pos(s(z1))) → COND_EVAL(greatereq_int(z0, pos(s(s(z1)))), z0, pos(s(z1)))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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 instantiating [LPAR04] the rule EVAL(z0, pos(s(z1))) → COND_EVAL(greatereq_int(z0, pos(s(s(z1)))), z0, pos(s(z1))) we obtained the following new rules [LPAR04]:

EVAL(z0, pos(s(s(z1)))) → COND_EVAL(greatereq_int(z0, pos(s(s(s(z1))))), z0, pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Instantiation

                                                                            ↳ QDP

                                                                              ↳ Instantiation

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                            ↳ QDP

                ↳ UsableRulesProof

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

COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, pos(s(s(z1))))
EVAL(z0, pos(s(s(z1)))) → COND_EVAL(greatereq_int(z0, pos(s(s(s(z1))))), z0, pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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 instantiating [LPAR04] the rule COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, pos(s(s(z1)))) we obtained the following new rules [LPAR04]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ UsableRulesProof

                                                            ↳ QDP

                                                              ↳ Rewriting

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ QReductionProof

                                                                        ↳ QDP

                                                                          ↳ Instantiation

                                                                            ↳ QDP

                                                                              ↳ Instantiation

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ Instantiation

                                                                                        ↳ QDP

                                            ↳ QDP

                ↳ UsableRulesProof

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

EVAL(z0, pos(s(s(z1)))) → COND_EVAL(greatereq_int(z0, pos(s(s(s(z1))))), z0, pos(s(s(z1))))
COND_EVAL(true, z0, pos(s(s(z1)))) → EVAL(z0, pos(s(s(s(z1)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                ↳ UsableRulesProof

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

COND_EVAL(true, y0, neg(x1)) → EVAL(y0, minus_nat(s(0), x1))
EVAL(y0, neg(x1)) → COND_EVAL(greatereq_int(y0, minus_nat(s(0), x1)), y0, neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                ↳ UsableRulesProof

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

COND_EVAL(true, y0, neg(x1)) → EVAL(y0, minus_nat(s(0), x1))
EVAL(y0, neg(x1)) → COND_EVAL(greatereq_int(y0, minus_nat(s(0), x1)), y0, neg(x1))

The TRS R consists of the following rules:

minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(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))
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].

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)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ MNOCProof

                                                      ↳ SemLabProof

                ↳ UsableRulesProof

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

COND_EVAL(true, y0, neg(x1)) → EVAL(y0, minus_nat(s(0), x1))
EVAL(y0, neg(x1)) → COND_EVAL(greatereq_int(y0, minus_nat(s(0), x1)), y0, neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ MNOCProof

                                                        ↳ QDP

                                                      ↳ SemLabProof

                ↳ UsableRulesProof

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

COND_EVAL(true, y0, neg(x1)) → EVAL(y0, minus_nat(s(0), x1))
EVAL(y0, neg(x1)) → COND_EVAL(greatereq_int(y0, minus_nat(s(0), x1)), y0, neg(x1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all (P,Q,R)-chains.
We found the following model for the rules of the TRS R. Interpretation over the domain with elements from 0 to 1.pos: 0
greatereq_int: 0
true: 0
neg: 0
EVAL: 0
COND_EVAL: 0
false: 0
s: 0
minus_nat: 0
0: 1
By semantic labelling [SEMLAB] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:

COND_EVAL.0-0-0(true., y0, neg.0(x1)) → EVAL.0-0(y0, minus_nat.0-0(s.1(0.), x1))
EVAL.0-0(y0, neg.0(x1)) → COND_EVAL.0-0-0(greatereq_int.0-0(y0, minus_nat.0-0(s.1(0.), x1)), y0, neg.0(x1))
EVAL.0-0(y0, neg.1(x1)) → COND_EVAL.0-0-0(greatereq_int.0-0(y0, minus_nat.0-1(s.1(0.), x1)), y0, neg.1(x1))
EVAL.1-0(y0, neg.0(x1)) → COND_EVAL.0-1-0(greatereq_int.1-0(y0, minus_nat.0-0(s.1(0.), x1)), y0, neg.0(x1))
EVAL.1-0(y0, neg.1(x1)) → COND_EVAL.0-1-0(greatereq_int.1-0(y0, minus_nat.0-1(s.1(0.), x1)), y0, neg.1(x1))
COND_EVAL.0-0-0(true., y0, neg.1(x1)) → EVAL.0-0(y0, minus_nat.0-1(s.1(0.), x1))
COND_EVAL.0-1-0(true., y0, neg.0(x1)) → EVAL.1-0(y0, minus_nat.0-0(s.1(0.), x1))
COND_EVAL.0-1-0(true., y0, neg.1(x1)) → EVAL.1-0(y0, minus_nat.0-1(s.1(0.), x1))

The TRS R consists of the following rules:

minus_nat.1-0(0., s.1(y)) → neg.0(s.1(y))
greatereq_int.0-0(neg.1(x), pos.0(s.0(y))) → false.
greatereq_int.0-0(neg.1(x), pos.0(s.1(y))) → false.
minus_nat.1-0(0., s.0(y)) → neg.0(s.0(y))
minus_nat.0-0(s.1(x), s.1(y)) → minus_nat.1-1(x, y)
greatereq_int.0-0(pos.1(0.), pos.0(s.0(y))) → false.
greatereq_int.0-0(pos.0(x), neg.0(y)) → true.
minus_nat.0-0(s.1(x), s.0(y)) → minus_nat.1-0(x, y)
greatereq_int.0-0(pos.0(s.1(x)), pos.0(s.1(y))) → greatereq_int.0-0(pos.1(x), pos.1(y))
greatereq_int.0-0(neg.0(s.1(x)), neg.0(s.1(y))) → greatereq_int.0-0(neg.1(x), neg.1(y))
greatereq_int.0-0(pos.1(0.), pos.0(s.1(y))) → false.
greatereq_int.0-0(neg.0(s.0(x)), pos.1(0.)) → false.
greatereq_int.0-0(pos.1(x), neg.1(y)) → true.
minus_nat.1-1(0., 0.) → pos.1(0.)
greatereq_int.0-0(neg.0(s.1(x)), pos.1(0.)) → false.
greatereq_int.0-0(neg.0(s.0(x)), neg.0(s.0(y))) → greatereq_int.0-0(neg.0(x), neg.0(y))
minus_nat.0-1(s.0(x), 0.) → pos.0(s.0(x))
greatereq_int.0-0(pos.0(s.1(x)), pos.0(s.0(y))) → greatereq_int.0-0(pos.1(x), pos.0(y))
greatereq_int.0-0(pos.0(s.0(x)), pos.0(s.1(y))) → greatereq_int.0-0(pos.0(x), pos.1(y))
greatereq_int.0-0(neg.0(s.0(x)), neg.1(0.)) → false.
greatereq_int.0-0(pos.1(x), neg.0(y)) → true.
greatereq_int.0-0(pos.0(x), neg.1(y)) → true.
greatereq_int.0-0(neg.1(0.), pos.1(0.)) → true.
greatereq_int.0-0(neg.0(x), pos.0(s.0(y))) → false.
greatereq_int.0-0(pos.0(s.0(x)), pos.0(s.0(y))) → greatereq_int.0-0(pos.0(x), pos.0(y))
greatereq_int.0-0(pos.1(x), pos.1(0.)) → true.
greatereq_int.0-0(neg.0(s.1(x)), neg.0(s.0(y))) → greatereq_int.0-0(neg.1(x), neg.0(y))
greatereq_int.0-0(neg.0(s.0(x)), neg.0(s.1(y))) → greatereq_int.0-0(neg.0(x), neg.1(y))
minus_nat.0-1(s.1(x), 0.) → pos.0(s.1(x))
minus_nat.0-0(s.0(x), s.0(y)) → minus_nat.0-0(x, y)
greatereq_int.0-0(pos.0(x), pos.1(0.)) → true.
greatereq_int.0-0(neg.0(s.1(x)), neg.1(0.)) → false.
greatereq_int.0-0(neg.1(0.), neg.1(y)) → true.
minus_nat.0-0(s.0(x), s.1(y)) → minus_nat.0-1(x, y)
greatereq_int.0-0(neg.1(0.), neg.0(y)) → true.
greatereq_int.0-0(neg.0(x), pos.0(s.1(y))) → false.

The set Q consists of the following terms:

minus_nat.1-1(0., 0.)
minus_nat.1-0(0., s.0(x0))
minus_nat.1-0(0., s.1(x0))
minus_nat.0-1(s.0(x0), 0.)
minus_nat.0-1(s.1(x0), 0.)
minus_nat.0-0(s.0(x0), s.0(x1))
minus_nat.0-0(s.0(x0), s.1(x1))
minus_nat.0-0(s.1(x0), s.0(x1))
minus_nat.0-0(s.1(x0), s.1(x1))
greatereq_int.0-0(pos.0(x0), pos.1(0.))
greatereq_int.0-0(pos.1(x0), pos.1(0.))
greatereq_int.0-0(neg.1(0.), pos.1(0.))
greatereq_int.0-0(neg.1(0.), neg.0(x0))
greatereq_int.0-0(neg.1(0.), neg.1(x0))
greatereq_int.0-0(pos.0(x0), neg.0(x1))
greatereq_int.0-0(pos.0(x0), neg.1(x1))
greatereq_int.0-0(pos.1(x0), neg.0(x1))
greatereq_int.0-0(pos.1(x0), neg.1(x1))
greatereq_int.0-0(pos.1(0.), pos.0(s.0(x0)))
greatereq_int.0-0(pos.1(0.), pos.0(s.1(x0)))
greatereq_int.0-0(neg.0(x0), pos.0(s.0(x1)))
greatereq_int.0-0(neg.0(x0), pos.0(s.1(x1)))
greatereq_int.0-0(neg.1(x0), pos.0(s.0(x1)))
greatereq_int.0-0(neg.1(x0), pos.0(s.1(x1)))
greatereq_int.0-0(neg.0(s.0(x0)), pos.1(0.))
greatereq_int.0-0(neg.0(s.1(x0)), pos.1(0.))
greatereq_int.0-0(neg.0(s.0(x0)), neg.1(0.))
greatereq_int.0-0(neg.0(s.1(x0)), neg.1(0.))
greatereq_int.0-0(pos.0(s.0(x0)), pos.0(s.0(x1)))
greatereq_int.0-0(pos.0(s.0(x0)), pos.0(s.1(x1)))
greatereq_int.0-0(pos.0(s.1(x0)), pos.0(s.0(x1)))
greatereq_int.0-0(pos.0(s.1(x0)), pos.0(s.1(x1)))
greatereq_int.0-0(neg.0(s.0(x0)), neg.0(s.0(x1)))
greatereq_int.0-0(neg.0(s.0(x0)), neg.0(s.1(x1)))
greatereq_int.0-0(neg.0(s.1(x0)), neg.0(s.0(x1)))
greatereq_int.0-0(neg.0(s.1(x0)), neg.0(s.1(x1)))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ MNOCProof

                                                      ↳ SemLabProof

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

COND_EVAL.0-0-0(true., y0, neg.0(x1)) → EVAL.0-0(y0, minus_nat.0-0(s.1(0.), x1))
EVAL.0-0(y0, neg.0(x1)) → COND_EVAL.0-0-0(greatereq_int.0-0(y0, minus_nat.0-0(s.1(0.), x1)), y0, neg.0(x1))
EVAL.0-0(y0, neg.1(x1)) → COND_EVAL.0-0-0(greatereq_int.0-0(y0, minus_nat.0-1(s.1(0.), x1)), y0, neg.1(x1))
EVAL.1-0(y0, neg.0(x1)) → COND_EVAL.0-1-0(greatereq_int.1-0(y0, minus_nat.0-0(s.1(0.), x1)), y0, neg.0(x1))
EVAL.1-0(y0, neg.1(x1)) → COND_EVAL.0-1-0(greatereq_int.1-0(y0, minus_nat.0-1(s.1(0.), x1)), y0, neg.1(x1))
COND_EVAL.0-0-0(true., y0, neg.1(x1)) → EVAL.0-0(y0, minus_nat.0-1(s.1(0.), x1))
COND_EVAL.0-1-0(true., y0, neg.0(x1)) → EVAL.1-0(y0, minus_nat.0-0(s.1(0.), x1))
COND_EVAL.0-1-0(true., y0, neg.1(x1)) → EVAL.1-0(y0, minus_nat.0-1(s.1(0.), x1))

The TRS R consists of the following rules:

minus_nat.1-0(0., s.1(y)) → neg.0(s.1(y))
greatereq_int.0-0(neg.1(x), pos.0(s.0(y))) → false.
greatereq_int.0-0(neg.1(x), pos.0(s.1(y))) → false.
minus_nat.1-0(0., s.0(y)) → neg.0(s.0(y))
minus_nat.0-0(s.1(x), s.1(y)) → minus_nat.1-1(x, y)
greatereq_int.0-0(pos.1(0.), pos.0(s.0(y))) → false.
greatereq_int.0-0(pos.0(x), neg.0(y)) → true.
minus_nat.0-0(s.1(x), s.0(y)) → minus_nat.1-0(x, y)
greatereq_int.0-0(pos.0(s.1(x)), pos.0(s.1(y))) → greatereq_int.0-0(pos.1(x), pos.1(y))
greatereq_int.0-0(neg.0(s.1(x)), neg.0(s.1(y))) → greatereq_int.0-0(neg.1(x), neg.1(y))
greatereq_int.0-0(pos.1(0.), pos.0(s.1(y))) → false.
greatereq_int.0-0(neg.0(s.0(x)), pos.1(0.)) → false.
greatereq_int.0-0(pos.1(x), neg.1(y)) → true.
minus_nat.1-1(0., 0.) → pos.1(0.)
greatereq_int.0-0(neg.0(s.1(x)), pos.1(0.)) → false.
greatereq_int.0-0(neg.0(s.0(x)), neg.0(s.0(y))) → greatereq_int.0-0(neg.0(x), neg.0(y))
minus_nat.0-1(s.0(x), 0.) → pos.0(s.0(x))
greatereq_int.0-0(pos.0(s.1(x)), pos.0(s.0(y))) → greatereq_int.0-0(pos.1(x), pos.0(y))
greatereq_int.0-0(pos.0(s.0(x)), pos.0(s.1(y))) → greatereq_int.0-0(pos.0(x), pos.1(y))
greatereq_int.0-0(neg.0(s.0(x)), neg.1(0.)) → false.
greatereq_int.0-0(pos.1(x), neg.0(y)) → true.
greatereq_int.0-0(pos.0(x), neg.1(y)) → true.
greatereq_int.0-0(neg.1(0.), pos.1(0.)) → true.
greatereq_int.0-0(neg.0(x), pos.0(s.0(y))) → false.
greatereq_int.0-0(pos.0(s.0(x)), pos.0(s.0(y))) → greatereq_int.0-0(pos.0(x), pos.0(y))
greatereq_int.0-0(pos.1(x), pos.1(0.)) → true.
greatereq_int.0-0(neg.0(s.1(x)), neg.0(s.0(y))) → greatereq_int.0-0(neg.1(x), neg.0(y))
greatereq_int.0-0(neg.0(s.0(x)), neg.0(s.1(y))) → greatereq_int.0-0(neg.0(x), neg.1(y))
minus_nat.0-1(s.1(x), 0.) → pos.0(s.1(x))
minus_nat.0-0(s.0(x), s.0(y)) → minus_nat.0-0(x, y)
greatereq_int.0-0(pos.0(x), pos.1(0.)) → true.
greatereq_int.0-0(neg.0(s.1(x)), neg.1(0.)) → false.
greatereq_int.0-0(neg.1(0.), neg.1(y)) → true.
minus_nat.0-0(s.0(x), s.1(y)) → minus_nat.0-1(x, y)
greatereq_int.0-0(neg.1(0.), neg.0(y)) → true.
greatereq_int.0-0(neg.0(x), pos.0(s.1(y))) → false.

The set Q consists of the following terms:

minus_nat.1-1(0., 0.)
minus_nat.1-0(0., s.0(x0))
minus_nat.1-0(0., s.1(x0))
minus_nat.0-1(s.0(x0), 0.)
minus_nat.0-1(s.1(x0), 0.)
minus_nat.0-0(s.0(x0), s.0(x1))
minus_nat.0-0(s.0(x0), s.1(x1))
minus_nat.0-0(s.1(x0), s.0(x1))
minus_nat.0-0(s.1(x0), s.1(x1))
greatereq_int.0-0(pos.0(x0), pos.1(0.))
greatereq_int.0-0(pos.1(x0), pos.1(0.))
greatereq_int.0-0(neg.1(0.), pos.1(0.))
greatereq_int.0-0(neg.1(0.), neg.0(x0))
greatereq_int.0-0(neg.1(0.), neg.1(x0))
greatereq_int.0-0(pos.0(x0), neg.0(x1))
greatereq_int.0-0(pos.0(x0), neg.1(x1))
greatereq_int.0-0(pos.1(x0), neg.0(x1))
greatereq_int.0-0(pos.1(x0), neg.1(x1))
greatereq_int.0-0(pos.1(0.), pos.0(s.0(x0)))
greatereq_int.0-0(pos.1(0.), pos.0(s.1(x0)))
greatereq_int.0-0(neg.0(x0), pos.0(s.0(x1)))
greatereq_int.0-0(neg.0(x0), pos.0(s.1(x1)))
greatereq_int.0-0(neg.1(x0), pos.0(s.0(x1)))
greatereq_int.0-0(neg.1(x0), pos.0(s.1(x1)))
greatereq_int.0-0(neg.0(s.0(x0)), pos.1(0.))
greatereq_int.0-0(neg.0(s.1(x0)), pos.1(0.))
greatereq_int.0-0(neg.0(s.0(x0)), neg.1(0.))
greatereq_int.0-0(neg.0(s.1(x0)), neg.1(0.))
greatereq_int.0-0(pos.0(s.0(x0)), pos.0(s.0(x1)))
greatereq_int.0-0(pos.0(s.0(x0)), pos.0(s.1(x1)))
greatereq_int.0-0(pos.0(s.1(x0)), pos.0(s.0(x1)))
greatereq_int.0-0(pos.0(s.1(x0)), pos.0(s.1(x1)))
greatereq_int.0-0(neg.0(s.0(x0)), neg.0(s.0(x1)))
greatereq_int.0-0(neg.0(s.0(x0)), neg.0(s.1(x1)))
greatereq_int.0-0(neg.0(s.1(x0)), neg.0(s.0(x1)))
greatereq_int.0-0(neg.0(s.1(x0)), neg.0(s.1(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ MNOCProof

                                                      ↳ SemLabProof

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

                ↳ UsableRulesProof

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

EVAL.0-0(y0, neg.0(x1)) → COND_EVAL.0-0-0(greatereq_int.0-0(y0, minus_nat.0-0(s.1(0.), x1)), y0, neg.0(x1))
COND_EVAL.0-0-0(true., y0, neg.0(x1)) → EVAL.0-0(y0, minus_nat.0-0(s.1(0.), x1))

The TRS R consists of the following rules:

minus_nat.1-0(0., s.1(y)) → neg.0(s.1(y))
greatereq_int.0-0(neg.1(x), pos.0(s.0(y))) → false.
greatereq_int.0-0(neg.1(x), pos.0(s.1(y))) → false.
minus_nat.1-0(0., s.0(y)) → neg.0(s.0(y))
minus_nat.0-0(s.1(x), s.1(y)) → minus_nat.1-1(x, y)
greatereq_int.0-0(pos.1(0.), pos.0(s.0(y))) → false.
greatereq_int.0-0(pos.0(x), neg.0(y)) → true.
minus_nat.0-0(s.1(x), s.0(y)) → minus_nat.1-0(x, y)
greatereq_int.0-0(pos.0(s.1(x)), pos.0(s.1(y))) → greatereq_int.0-0(pos.1(x), pos.1(y))
greatereq_int.0-0(neg.0(s.1(x)), neg.0(s.1(y))) → greatereq_int.0-0(neg.1(x), neg.1(y))
greatereq_int.0-0(pos.1(0.), pos.0(s.1(y))) → false.
greatereq_int.0-0(neg.0(s.0(x)), pos.1(0.)) → false.
greatereq_int.0-0(pos.1(x), neg.1(y)) → true.
minus_nat.1-1(0., 0.) → pos.1(0.)
greatereq_int.0-0(neg.0(s.1(x)), pos.1(0.)) → false.
greatereq_int.0-0(neg.0(s.0(x)), neg.0(s.0(y))) → greatereq_int.0-0(neg.0(x), neg.0(y))
minus_nat.0-1(s.0(x), 0.) → pos.0(s.0(x))
greatereq_int.0-0(pos.0(s.1(x)), pos.0(s.0(y))) → greatereq_int.0-0(pos.1(x), pos.0(y))
greatereq_int.0-0(pos.0(s.0(x)), pos.0(s.1(y))) → greatereq_int.0-0(pos.0(x), pos.1(y))
greatereq_int.0-0(neg.0(s.0(x)), neg.1(0.)) → false.
greatereq_int.0-0(pos.1(x), neg.0(y)) → true.
greatereq_int.0-0(pos.0(x), neg.1(y)) → true.
greatereq_int.0-0(neg.1(0.), pos.1(0.)) → true.
greatereq_int.0-0(neg.0(x), pos.0(s.0(y))) → false.
greatereq_int.0-0(pos.0(s.0(x)), pos.0(s.0(y))) → greatereq_int.0-0(pos.0(x), pos.0(y))
greatereq_int.0-0(pos.1(x), pos.1(0.)) → true.
greatereq_int.0-0(neg.0(s.1(x)), neg.0(s.0(y))) → greatereq_int.0-0(neg.1(x), neg.0(y))
greatereq_int.0-0(neg.0(s.0(x)), neg.0(s.1(y))) → greatereq_int.0-0(neg.0(x), neg.1(y))
minus_nat.0-1(s.1(x), 0.) → pos.0(s.1(x))
minus_nat.0-0(s.0(x), s.0(y)) → minus_nat.0-0(x, y)
greatereq_int.0-0(pos.0(x), pos.1(0.)) → true.
greatereq_int.0-0(neg.0(s.1(x)), neg.1(0.)) → false.
greatereq_int.0-0(neg.1(0.), neg.1(y)) → true.
minus_nat.0-0(s.0(x), s.1(y)) → minus_nat.0-1(x, y)
greatereq_int.0-0(neg.1(0.), neg.0(y)) → true.
greatereq_int.0-0(neg.0(x), pos.0(s.1(y))) → false.

The set Q consists of the following terms:

minus_nat.1-1(0., 0.)
minus_nat.1-0(0., s.0(x0))
minus_nat.1-0(0., s.1(x0))
minus_nat.0-1(s.0(x0), 0.)
minus_nat.0-1(s.1(x0), 0.)
minus_nat.0-0(s.0(x0), s.0(x1))
minus_nat.0-0(s.0(x0), s.1(x1))
minus_nat.0-0(s.1(x0), s.0(x1))
minus_nat.0-0(s.1(x0), s.1(x1))
greatereq_int.0-0(pos.0(x0), pos.1(0.))
greatereq_int.0-0(pos.1(x0), pos.1(0.))
greatereq_int.0-0(neg.1(0.), pos.1(0.))
greatereq_int.0-0(neg.1(0.), neg.0(x0))
greatereq_int.0-0(neg.1(0.), neg.1(x0))
greatereq_int.0-0(pos.0(x0), neg.0(x1))
greatereq_int.0-0(pos.0(x0), neg.1(x1))
greatereq_int.0-0(pos.1(x0), neg.0(x1))
greatereq_int.0-0(pos.1(x0), neg.1(x1))
greatereq_int.0-0(pos.1(0.), pos.0(s.0(x0)))
greatereq_int.0-0(pos.1(0.), pos.0(s.1(x0)))
greatereq_int.0-0(neg.0(x0), pos.0(s.0(x1)))
greatereq_int.0-0(neg.0(x0), pos.0(s.1(x1)))
greatereq_int.0-0(neg.1(x0), pos.0(s.0(x1)))
greatereq_int.0-0(neg.1(x0), pos.0(s.1(x1)))
greatereq_int.0-0(neg.0(s.0(x0)), pos.1(0.))
greatereq_int.0-0(neg.0(s.1(x0)), pos.1(0.))
greatereq_int.0-0(neg.0(s.0(x0)), neg.1(0.))
greatereq_int.0-0(neg.0(s.1(x0)), neg.1(0.))
greatereq_int.0-0(pos.0(s.0(x0)), pos.0(s.0(x1)))
greatereq_int.0-0(pos.0(s.0(x0)), pos.0(s.1(x1)))
greatereq_int.0-0(pos.0(s.1(x0)), pos.0(s.0(x1)))
greatereq_int.0-0(pos.0(s.1(x0)), pos.0(s.1(x1)))
greatereq_int.0-0(neg.0(s.0(x0)), neg.0(s.0(x1)))
greatereq_int.0-0(neg.0(s.0(x0)), neg.0(s.1(x1)))
greatereq_int.0-0(neg.0(s.1(x0)), neg.0(s.0(x1)))
greatereq_int.0-0(neg.0(s.1(x0)), neg.0(s.1(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.0-0(y0, neg.0(x1)) → COND_EVAL.0-0-0(greatereq_int.0-0(y0, minus_nat.0-0(s.1(0.), x1)), y0, neg.0(x1))

The remaining pairs can at least be oriented weakly.

COND_EVAL.0-0-0(true., y0, neg.0(x1)) → EVAL.0-0(y0, minus_nat.0-0(s.1(0.), x1))

Used ordering: Polynomial interpretation [POLO]:

POL(0.) = 0
POL(COND_EVAL.0-0-0(x₁, x₂, x₃)) = x₃
POL(EVAL.0-0(x₁, x₂)) = 1 + x₂
POL(false.) = 0
POL(greatereq_int.0-0(x₁, x₂)) = 0
POL(minus_nat.0-0(x₁, x₂)) = x₂
POL(minus_nat.1-0(x₁, x₂)) = 1 + x₂
POL(minus_nat.1-1(x₁, x₂)) = 0
POL(neg.0(x₁)) = 1 + x₁
POL(neg.1(x₁)) = 0
POL(pos.0(x₁)) = 0
POL(pos.1(x₁)) = 0
POL(s.0(x₁)) = 1 + x₁
POL(s.1(x₁)) = 0
POL(true.) = 0

The following usable rules [FROCOS05] were oriented:

minus_nat.0-0(s.1(x), s.1(y)) → minus_nat.1-1(x, y)
minus_nat.1-0(0., s.0(y)) → neg.0(s.0(y))
minus_nat.1-0(0., s.1(y)) → neg.0(s.1(y))
minus_nat.1-1(0., 0.) → pos.1(0.)
minus_nat.0-0(s.1(x), s.0(y)) → minus_nat.1-0(x, y)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ MNOCProof

                                                      ↳ SemLabProof

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

COND_EVAL.0-0-0(true., y0, neg.0(x1)) → EVAL.0-0(y0, minus_nat.0-0(s.1(0.), x1))

The TRS R consists of the following rules:

minus_nat.1-0(0., s.1(y)) → neg.0(s.1(y))
greatereq_int.0-0(neg.1(x), pos.0(s.0(y))) → false.
greatereq_int.0-0(neg.1(x), pos.0(s.1(y))) → false.
minus_nat.1-0(0., s.0(y)) → neg.0(s.0(y))
minus_nat.0-0(s.1(x), s.1(y)) → minus_nat.1-1(x, y)
greatereq_int.0-0(pos.1(0.), pos.0(s.0(y))) → false.
greatereq_int.0-0(pos.0(x), neg.0(y)) → true.
minus_nat.0-0(s.1(x), s.0(y)) → minus_nat.1-0(x, y)
greatereq_int.0-0(pos.0(s.1(x)), pos.0(s.1(y))) → greatereq_int.0-0(pos.1(x), pos.1(y))
greatereq_int.0-0(neg.0(s.1(x)), neg.0(s.1(y))) → greatereq_int.0-0(neg.1(x), neg.1(y))
greatereq_int.0-0(pos.1(0.), pos.0(s.1(y))) → false.
greatereq_int.0-0(neg.0(s.0(x)), pos.1(0.)) → false.
greatereq_int.0-0(pos.1(x), neg.1(y)) → true.
minus_nat.1-1(0., 0.) → pos.1(0.)
greatereq_int.0-0(neg.0(s.1(x)), pos.1(0.)) → false.
greatereq_int.0-0(neg.0(s.0(x)), neg.0(s.0(y))) → greatereq_int.0-0(neg.0(x), neg.0(y))
minus_nat.0-1(s.0(x), 0.) → pos.0(s.0(x))
greatereq_int.0-0(pos.0(s.1(x)), pos.0(s.0(y))) → greatereq_int.0-0(pos.1(x), pos.0(y))
greatereq_int.0-0(pos.0(s.0(x)), pos.0(s.1(y))) → greatereq_int.0-0(pos.0(x), pos.1(y))
greatereq_int.0-0(neg.0(s.0(x)), neg.1(0.)) → false.
greatereq_int.0-0(pos.1(x), neg.0(y)) → true.
greatereq_int.0-0(pos.0(x), neg.1(y)) → true.
greatereq_int.0-0(neg.1(0.), pos.1(0.)) → true.
greatereq_int.0-0(neg.0(x), pos.0(s.0(y))) → false.
greatereq_int.0-0(pos.0(s.0(x)), pos.0(s.0(y))) → greatereq_int.0-0(pos.0(x), pos.0(y))
greatereq_int.0-0(pos.1(x), pos.1(0.)) → true.
greatereq_int.0-0(neg.0(s.1(x)), neg.0(s.0(y))) → greatereq_int.0-0(neg.1(x), neg.0(y))
greatereq_int.0-0(neg.0(s.0(x)), neg.0(s.1(y))) → greatereq_int.0-0(neg.0(x), neg.1(y))
minus_nat.0-1(s.1(x), 0.) → pos.0(s.1(x))
minus_nat.0-0(s.0(x), s.0(y)) → minus_nat.0-0(x, y)
greatereq_int.0-0(pos.0(x), pos.1(0.)) → true.
greatereq_int.0-0(neg.0(s.1(x)), neg.1(0.)) → false.
greatereq_int.0-0(neg.1(0.), neg.1(y)) → true.
minus_nat.0-0(s.0(x), s.1(y)) → minus_nat.0-1(x, y)
greatereq_int.0-0(neg.1(0.), neg.0(y)) → true.
greatereq_int.0-0(neg.0(x), pos.0(s.1(y))) → false.

The set Q consists of the following terms:

minus_nat.1-1(0., 0.)
minus_nat.1-0(0., s.0(x0))
minus_nat.1-0(0., s.1(x0))
minus_nat.0-1(s.0(x0), 0.)
minus_nat.0-1(s.1(x0), 0.)
minus_nat.0-0(s.0(x0), s.0(x1))
minus_nat.0-0(s.0(x0), s.1(x1))
minus_nat.0-0(s.1(x0), s.0(x1))
minus_nat.0-0(s.1(x0), s.1(x1))
greatereq_int.0-0(pos.0(x0), pos.1(0.))
greatereq_int.0-0(pos.1(x0), pos.1(0.))
greatereq_int.0-0(neg.1(0.), pos.1(0.))
greatereq_int.0-0(neg.1(0.), neg.0(x0))
greatereq_int.0-0(neg.1(0.), neg.1(x0))
greatereq_int.0-0(pos.0(x0), neg.0(x1))
greatereq_int.0-0(pos.0(x0), neg.1(x1))
greatereq_int.0-0(pos.1(x0), neg.0(x1))
greatereq_int.0-0(pos.1(x0), neg.1(x1))
greatereq_int.0-0(pos.1(0.), pos.0(s.0(x0)))
greatereq_int.0-0(pos.1(0.), pos.0(s.1(x0)))
greatereq_int.0-0(neg.0(x0), pos.0(s.0(x1)))
greatereq_int.0-0(neg.0(x0), pos.0(s.1(x1)))
greatereq_int.0-0(neg.1(x0), pos.0(s.0(x1)))
greatereq_int.0-0(neg.1(x0), pos.0(s.1(x1)))
greatereq_int.0-0(neg.0(s.0(x0)), pos.1(0.))
greatereq_int.0-0(neg.0(s.1(x0)), pos.1(0.))
greatereq_int.0-0(neg.0(s.0(x0)), neg.1(0.))
greatereq_int.0-0(neg.0(s.1(x0)), neg.1(0.))
greatereq_int.0-0(pos.0(s.0(x0)), pos.0(s.0(x1)))
greatereq_int.0-0(pos.0(s.0(x0)), pos.0(s.1(x1)))
greatereq_int.0-0(pos.0(s.1(x0)), pos.0(s.0(x1)))
greatereq_int.0-0(pos.0(s.1(x0)), pos.0(s.1(x1)))
greatereq_int.0-0(neg.0(s.0(x0)), neg.0(s.0(x1)))
greatereq_int.0-0(neg.0(s.0(x0)), neg.0(s.1(x1)))
greatereq_int.0-0(neg.0(s.1(x0)), neg.0(s.0(x1)))
greatereq_int.0-0(neg.0(s.1(x0)), neg.0(s.1(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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