Termination proof of /tmp/tmpXd262X/increase2.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_f(TRUE, x, y, z) → f(x, y, +@z(z, 1@z))
f(x, y, z) → Cond_f(>@z(x, +@z(y, z)), x, y, z)
Cond_f1(TRUE, x, y, z) → f(x, +@z(y, 1@z), z)
f(x, y, z) → Cond_f1(>@z(x, +@z(y, z)), x, y, z)

The set Q consists of the following terms:

Cond_f(TRUE, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(TRUE, x0, x1, x2)

Represented integers and predefined function symbols by Terms

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

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

Cond_f(true, x, y, z) → f(x, y, plus_int(pos(s(0)), z))
f(x, y, z) → Cond_f(greater_int(x, plus_int(y, z)), x, y, z)
Cond_f1(true, x, y, z) → f(x, plus_int(pos(s(0)), y), z)
f(x, y, z) → Cond_f1(greater_int(x, plus_int(y, z)), x, y, z)
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)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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_F(true, x, y, z) → F(x, y, plus_int(pos(s(0)), z))
COND_F(true, x, y, z) → PLUS_INT(pos(s(0)), z)
F(x, y, z) → COND_F(greater_int(x, plus_int(y, z)), x, y, z)
F(x, y, z) → GREATER_INT(x, plus_int(y, z))
F(x, y, z) → PLUS_INT(y, z)
COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z)
COND_F1(true, x, y, z) → PLUS_INT(pos(s(0)), y)
F(x, y, z) → COND_F1(greater_int(x, plus_int(y, z)), x, y, z)
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)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

The TRS R consists of the following rules:

Cond_f(true, x, y, z) → f(x, y, plus_int(pos(s(0)), z))
f(x, y, z) → Cond_f(greater_int(x, plus_int(y, z)), x, y, z)
Cond_f1(true, x, y, z) → f(x, plus_int(pos(s(0)), y), z)
f(x, y, z) → Cond_f1(greater_int(x, plus_int(y, z)), x, y, z)
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)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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_F(true, x, y, z) → F(x, y, plus_int(pos(s(0)), z))
COND_F(true, x, y, z) → PLUS_INT(pos(s(0)), z)
F(x, y, z) → COND_F(greater_int(x, plus_int(y, z)), x, y, z)
F(x, y, z) → GREATER_INT(x, plus_int(y, z))
F(x, y, z) → PLUS_INT(y, z)
COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z)
COND_F1(true, x, y, z) → PLUS_INT(pos(s(0)), y)
F(x, y, z) → COND_F1(greater_int(x, plus_int(y, z)), x, y, z)
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)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

The TRS R consists of the following rules:

Cond_f(true, x, y, z) → f(x, y, plus_int(pos(s(0)), z))
f(x, y, z) → Cond_f(greater_int(x, plus_int(y, z)), x, y, z)
Cond_f1(true, x, y, z) → f(x, plus_int(pos(s(0)), y), z)
f(x, y, z) → Cond_f1(greater_int(x, plus_int(y, z)), x, y, z)
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)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 8 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:

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

The TRS R consists of the following rules:

Cond_f(true, x, y, z) → f(x, y, plus_int(pos(s(0)), z))
f(x, y, z) → Cond_f(greater_int(x, plus_int(y, z)), x, y, z)
Cond_f1(true, x, y, z) → f(x, plus_int(pos(s(0)), y), z)
f(x, y, z) → Cond_f1(greater_int(x, plus_int(y, z)), x, y, z)
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)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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:

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

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

Cond_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

The following dependency pairs can be deleted:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ 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:

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

The TRS R consists of the following rules:

Cond_f(true, x, y, z) → f(x, y, plus_int(pos(s(0)), z))
f(x, y, z) → Cond_f(greater_int(x, plus_int(y, z)), x, y, z)
Cond_f1(true, x, y, z) → f(x, plus_int(pos(s(0)), y), z)
f(x, y, z) → Cond_f1(greater_int(x, plus_int(y, z)), x, y, z)
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)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

Cond_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ 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_f(true, x, y, z) → f(x, y, plus_int(pos(s(0)), z))
f(x, y, z) → Cond_f(greater_int(x, plus_int(y, z)), x, y, z)
Cond_f1(true, x, y, z) → f(x, plus_int(pos(s(0)), y), z)
f(x, y, z) → Cond_f1(greater_int(x, plus_int(y, z)), x, y, z)
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)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ 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_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ 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_f(true, x, y, z) → f(x, y, plus_int(pos(s(0)), z))
f(x, y, z) → Cond_f(greater_int(x, plus_int(y, z)), x, y, z)
Cond_f1(true, x, y, z) → f(x, plus_int(pos(s(0)), y), z)
f(x, y, z) → Cond_f1(greater_int(x, plus_int(y, z)), x, y, z)
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)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ 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_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ 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

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

F(x, y, z) → COND_F(greater_int(x, plus_int(y, z)), x, y, z)
COND_F(true, x, y, z) → F(x, y, plus_int(pos(s(0)), z))
F(x, y, z) → COND_F1(greater_int(x, plus_int(y, z)), x, y, z)
COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z)

The TRS R consists of the following rules:

Cond_f(true, x, y, z) → f(x, y, plus_int(pos(s(0)), z))
f(x, y, z) → Cond_f(greater_int(x, plus_int(y, z)), x, y, z)
Cond_f1(true, x, y, z) → f(x, plus_int(pos(s(0)), y), z)
f(x, y, z) → Cond_f1(greater_int(x, plus_int(y, z)), x, y, z)
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)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

F(x, y, z) → COND_F(greater_int(x, plus_int(y, z)), x, y, z)
COND_F(true, x, y, z) → F(x, y, plus_int(pos(s(0)), z))
F(x, y, z) → COND_F1(greater_int(x, plus_int(y, z)), x, y, z)
COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z)

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
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_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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_f(true, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(true, x0, x1, x2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

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

F(x, y, z) → COND_F(greater_int(x, plus_int(y, z)), x, y, z)
COND_F(true, x, y, z) → F(x, y, plus_int(pos(s(0)), z))
F(x, y, z) → COND_F1(greater_int(x, plus_int(y, z)), x, y, z)
COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z)

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
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))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

[2,0]

Pair: COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z)
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

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

F(x, y, z, x_removed) → COND_F(greater_int(x, plus_int(y, z)), x, y, z, x_removed)
COND_F(true, x, y, z, x_removed) → F(x, y, plus_int(x_removed, z), x_removed)
F(x, y, z, x_removed) → COND_F1(greater_int(x, plus_int(y, z)), x, y, z, x_removed)
COND_F1(true, x, y, z, x_removed) → F(x, plus_int(x_removed, y), z, x_removed)

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
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))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

[2,0]

Pair: COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z)
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

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

F(x, y, z, x_removed) → COND_F(greater_int(x, plus_int(y, z)), x, y, z, x_removed)
COND_F(true, x, y, z, x_removed) → F(x, y, plus_int(x_removed, z), x_removed)
F(x, y, z, x_removed) → COND_F1(greater_int(x, plus_int(y, z)), x, y, z, x_removed)
COND_F1(true, x, y, z, x_removed) → F(x, plus_int(x_removed, y), z, x_removed)

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
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))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

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

COND_F(true, x, y, z) → F(x, y, plus_int(pos(s(0)), z))
F(x, y, z) → COND_F1(greater_int(x, plus_int(y, z)), x, y, z)
COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z)
F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
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))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule F(x, y, z) → COND_F1(greater_int(x, plus_int(y, z)), x, y, z) at position [0] we obtained the following new rules [LPAR04]:

F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

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

COND_F(true, x, y, z) → F(x, y, plus_int(pos(s(0)), z))
COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z)
F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
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))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

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

COND_F(true, x, y, z) → F(x, y, plus_int(pos(s(0)), z))
COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z)
F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_F(true, x, y, z) → F(x, y, plus_int(pos(s(0)), z)) at position [2] we obtained the following new rules [LPAR04]:

COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(plus_nat(s(0), x1)))
COND_F(true, y0, y1, neg(x1)) → F(y0, y1, 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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

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

COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z)
F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(plus_nat(s(0), x1)))
COND_F(true, y0, y1, neg(x1)) → F(y0, y1, minus_nat(s(0), x1))

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(plus_nat(s(0), x1))) at position [2,0] we obtained the following new rules [LPAR04]:

COND_F(true, y0, y1, pos(x1)) → F(y0, y1, 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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

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

COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z)
F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F(true, y0, y1, neg(x1)) → F(y0, y1, minus_nat(s(0), x1))
COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(s(plus_nat(0, x1))))

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(s(plus_nat(0, x1)))) at position [2,0,0] we obtained the following new rules [LPAR04]:

COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

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

COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z)
F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F(true, y0, y1, neg(x1)) → F(y0, y1, minus_nat(s(0), x1))
COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(s(x1)))

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_F1(true, x, y, z) → F(x, plus_int(pos(s(0)), y), z) at position [1] we obtained the following new rules [LPAR04]:

COND_F1(true, y0, pos(x1), y2) → F(y0, pos(plus_nat(s(0), x1)), y2)
COND_F1(true, y0, neg(x1), y2) → F(y0, minus_nat(s(0), x1), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

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

F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F(true, y0, y1, neg(x1)) → F(y0, y1, minus_nat(s(0), x1))
COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(s(x1)))
COND_F1(true, y0, pos(x1), y2) → F(y0, pos(plus_nat(s(0), x1)), y2)
COND_F1(true, y0, neg(x1), y2) → F(y0, minus_nat(s(0), x1), y2)

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

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

F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F(true, y0, y1, neg(x1)) → F(y0, y1, minus_nat(s(0), x1))
COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(s(x1)))
COND_F1(true, y0, pos(x1), y2) → F(y0, pos(plus_nat(s(0), x1)), y2)
COND_F1(true, y0, neg(x1), y2) → F(y0, minus_nat(s(0), x1), y2)

The TRS R consists of the following rules:

plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

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

F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F(true, y0, y1, neg(x1)) → F(y0, y1, minus_nat(s(0), x1))
COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(s(x1)))
COND_F1(true, y0, pos(x1), y2) → F(y0, pos(plus_nat(s(0), x1)), y2)
COND_F1(true, y0, neg(x1), y2) → F(y0, minus_nat(s(0), x1), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_F1(true, y0, pos(x1), y2) → F(y0, pos(plus_nat(s(0), x1)), y2) at position [1,0] we obtained the following new rules [LPAR04]:

COND_F1(true, y0, pos(x1), y2) → F(y0, pos(s(plus_nat(0, x1))), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

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

F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F(true, y0, y1, neg(x1)) → F(y0, y1, minus_nat(s(0), x1))
COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(s(x1)))
COND_F1(true, y0, neg(x1), y2) → F(y0, minus_nat(s(0), x1), y2)
COND_F1(true, y0, pos(x1), y2) → F(y0, pos(s(plus_nat(0, x1))), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_F1(true, y0, pos(x1), y2) → F(y0, pos(s(plus_nat(0, x1))), y2) at position [1,0,0] we obtained the following new rules [LPAR04]:

COND_F1(true, y0, pos(x1), y2) → F(y0, pos(s(x1)), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

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

F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F(true, y0, y1, neg(x1)) → F(y0, y1, minus_nat(s(0), x1))
COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(s(x1)))
COND_F1(true, y0, neg(x1), y2) → F(y0, minus_nat(s(0), x1), y2)
COND_F1(true, y0, pos(x1), y2) → F(y0, pos(s(x1)), y2)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F(true, y0, y1, neg(x1)) → F(y0, y1, minus_nat(s(0), x1)) we obtained the following new rules [LPAR04]:

COND_F(true, z0, neg(z1), neg(z2)) → F(z0, neg(z1), minus_nat(s(0), z2))
COND_F(true, z0, pos(z1), neg(z2)) → F(z0, pos(z1), minus_nat(s(0), z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

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

F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(s(x1)))
COND_F1(true, y0, neg(x1), y2) → F(y0, minus_nat(s(0), x1), y2)
COND_F1(true, y0, pos(x1), y2) → F(y0, pos(s(x1)), y2)
COND_F(true, z0, neg(z1), neg(z2)) → F(z0, neg(z1), minus_nat(s(0), z2))
COND_F(true, z0, pos(z1), neg(z2)) → F(z0, pos(z1), minus_nat(s(0), z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F(true, y0, y1, pos(x1)) → F(y0, y1, pos(s(x1))) we obtained the following new rules [LPAR04]:

COND_F(true, z0, pos(z1), pos(z2)) → F(z0, pos(z1), pos(s(z2)))
COND_F(true, z0, neg(z1), pos(z2)) → F(z0, neg(z1), pos(s(z2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

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

F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F1(true, y0, neg(x1), y2) → F(y0, minus_nat(s(0), x1), y2)
COND_F1(true, y0, pos(x1), y2) → F(y0, pos(s(x1)), y2)
COND_F(true, z0, neg(z1), neg(z2)) → F(z0, neg(z1), minus_nat(s(0), z2))
COND_F(true, z0, pos(z1), neg(z2)) → F(z0, pos(z1), minus_nat(s(0), z2))
COND_F(true, z0, pos(z1), pos(z2)) → F(z0, pos(z1), pos(s(z2)))
COND_F(true, z0, neg(z1), pos(z2)) → F(z0, neg(z1), pos(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                ↳ QDP

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

COND_F(true, z0, pos(z1), neg(z2)) → F(z0, pos(z1), minus_nat(s(0), z2))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F(true, z0, pos(z1), pos(z2)) → F(z0, pos(z1), pos(s(z2)))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F1(true, y0, pos(x1), y2) → F(y0, pos(s(x1)), y2)
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F1(true, y0, pos(x1), y2) → F(y0, pos(s(x1)), y2) we obtained the following new rules [LPAR04]:

COND_F1(true, z0, pos(z1), neg(z2)) → F(z0, pos(s(z1)), neg(z2))
COND_F1(true, z0, pos(z1), pos(z2)) → F(z0, pos(s(z1)), pos(z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                ↳ QDP

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

COND_F(true, z0, pos(z1), neg(z2)) → F(z0, pos(z1), minus_nat(s(0), z2))
F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F(true, z0, pos(z1), pos(z2)) → F(z0, pos(z1), pos(s(z2)))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
COND_F1(true, z0, pos(z1), neg(z2)) → F(z0, pos(s(z1)), neg(z2))
COND_F1(true, z0, pos(z1), pos(z2)) → F(z0, pos(s(z1)), pos(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F(true, z0, pos(z1), pos(z2)) → F(z0, pos(z1), pos(s(z2)))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F1(true, z0, pos(z1), pos(z2)) → F(z0, pos(s(z1)), pos(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F(true, z0, pos(z1), pos(z2)) → F(z0, pos(z1), pos(s(z2)))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F1(true, z0, pos(z1), pos(z2)) → F(z0, pos(s(z1)), pos(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

minus_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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F(true, z0, pos(z1), pos(z2)) → F(z0, pos(z1), pos(s(z2)))
F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F1(true, z0, pos(z1), pos(z2)) → F(z0, pos(s(z1)), pos(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule F(y0, pos(x0), pos(x1)) → COND_F(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1)) we obtained the following new rules [LPAR04]:

F(z0, pos(z1), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(z2)))), z0, pos(z1), pos(s(z2)))
F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(plus_nat(s(z1), z2))), z0, pos(s(z1)), pos(z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F(true, z0, pos(z1), pos(z2)) → F(z0, pos(z1), pos(s(z2)))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F1(true, z0, pos(z1), pos(z2)) → F(z0, pos(s(z1)), pos(z2))
F(z0, pos(z1), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(z2)))), z0, pos(z1), pos(s(z2)))
F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(plus_nat(s(z1), z2))), z0, pos(s(z1)), pos(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F(true, z0, pos(z1), pos(z2)) → F(z0, pos(z1), pos(s(z2)))
F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F1(true, z0, pos(z1), pos(z2)) → F(z0, pos(s(z1)), pos(z2))
F(z0, pos(z1), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(z2)))), z0, pos(z1), pos(s(z2)))
F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F(true, z0, pos(z1), pos(z2)) → F(z0, pos(z1), pos(s(z2))) we obtained the following new rules [LPAR04]:

COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1))
COND_F1(true, z0, pos(z1), pos(z2)) → F(z0, pos(s(z1)), pos(z2))
F(z0, pos(z1), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(z2)))), z0, pos(z1), pos(s(z2)))
F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule F(y0, pos(x0), pos(x1)) → COND_F1(greater_int(y0, pos(plus_nat(x0, x1))), y0, pos(x0), pos(x1)) we obtained the following new rules [LPAR04]:

F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(plus_nat(s(z1), z2))), z0, pos(s(z1)), pos(z2))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(plus_nat(s(z1), s(z2)))), z0, pos(s(z1)), pos(s(z2)))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F1(true, z0, pos(z1), pos(z2)) → F(z0, pos(s(z1)), pos(z2))
F(z0, pos(z1), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(z2)))), z0, pos(z1), pos(s(z2)))
F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(plus_nat(s(z1), z2))), z0, pos(s(z1)), pos(z2))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(plus_nat(s(z1), s(z2)))), z0, pos(s(z1)), pos(s(z2)))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(plus_nat(s(z1), z2))), z0, pos(s(z1)), pos(z2)) at position [0,1,0] we obtained the following new rules [LPAR04]:

F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F1(true, z0, pos(z1), pos(z2)) → F(z0, pos(s(z1)), pos(z2))
F(z0, pos(z1), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(z2)))), z0, pos(z1), pos(s(z2)))
F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(plus_nat(s(z1), s(z2)))), z0, pos(s(z1)), pos(s(z2)))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(plus_nat(s(z1), s(z2)))), z0, pos(s(z1)), pos(s(z2))) at position [0,1,0] we obtained the following new rules [LPAR04]:

F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F1(true, z0, pos(z1), pos(z2)) → F(z0, pos(s(z1)), pos(z2))
F(z0, pos(z1), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(z2)))), z0, pos(z1), pos(s(z2)))
F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F1(true, z0, pos(z1), pos(z2)) → F(z0, pos(s(z1)), pos(z2)) we obtained the following new rules [LPAR04]:

COND_F1(true, z0, pos(s(z1)), pos(s(z2))) → F(z0, pos(s(s(z1))), pos(s(z2)))
COND_F1(true, z0, pos(z1), pos(s(s(z2)))) → F(z0, pos(s(z1)), pos(s(s(z2))))
COND_F1(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(s(z1))), pos(z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Instantiation

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

F(z0, pos(z1), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(z2)))), z0, pos(z1), pos(s(z2)))
F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
COND_F1(true, z0, pos(s(z1)), pos(s(z2))) → F(z0, pos(s(s(z1))), pos(s(z2)))
COND_F1(true, z0, pos(z1), pos(s(s(z2)))) → F(z0, pos(s(z1)), pos(s(s(z2))))
COND_F1(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(s(z1))), pos(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule F(z0, pos(z1), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(z2)))), z0, pos(z1), pos(s(z2))) we obtained the following new rules [LPAR04]:

F(z0, pos(z1), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(s(z1), s(s(z2))))), z0, pos(s(z1)), pos(s(s(z2))))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(s(s(z1)), s(z2)))), z0, pos(s(s(z1))), pos(s(z2)))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(s(z1), s(z2)))), z0, pos(s(z1)), pos(s(z2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Instantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
COND_F1(true, z0, pos(s(z1)), pos(s(z2))) → F(z0, pos(s(s(z1))), pos(s(z2)))
COND_F1(true, z0, pos(z1), pos(s(s(z2)))) → F(z0, pos(s(z1)), pos(s(s(z2))))
COND_F1(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(s(z1))), pos(z2))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(s(z1), s(s(z2))))), z0, pos(s(z1)), pos(s(s(z2))))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(s(s(z1)), s(z2)))), z0, pos(s(s(z1))), pos(s(z2)))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(s(z1), s(z2)))), z0, pos(s(z1)), pos(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

F(z0, pos(s(z1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(s(z2)))))), z0, pos(s(z1)), pos(s(s(z2))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Instantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
COND_F1(true, z0, pos(s(z1)), pos(s(z2))) → F(z0, pos(s(s(z1))), pos(s(z2)))
COND_F1(true, z0, pos(z1), pos(s(s(z2)))) → F(z0, pos(s(z1)), pos(s(s(z2))))
COND_F1(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(s(z1))), pos(z2))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(s(s(z1)), s(z2)))), z0, pos(s(s(z1))), pos(s(z2)))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(s(z1), s(z2)))), z0, pos(s(z1)), pos(s(z2)))
F(z0, pos(s(z1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(s(z2)))))), z0, pos(s(z1)), pos(s(s(z2))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(s(z1), s(z2))))), z0, pos(s(s(z1))), pos(s(z2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Instantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
COND_F1(true, z0, pos(s(z1)), pos(s(z2))) → F(z0, pos(s(s(z1))), pos(s(z2)))
COND_F1(true, z0, pos(z1), pos(s(s(z2)))) → F(z0, pos(s(z1)), pos(s(s(z2))))
COND_F1(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(s(z1))), pos(z2))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(plus_nat(s(z1), s(z2)))), z0, pos(s(z1)), pos(s(z2)))
F(z0, pos(s(z1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(s(z2)))))), z0, pos(s(z1)), pos(s(s(z2))))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(s(z1), s(z2))))), z0, pos(s(s(z1))), pos(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Instantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
COND_F1(true, z0, pos(s(z1)), pos(s(z2))) → F(z0, pos(s(s(z1))), pos(s(z2)))
COND_F1(true, z0, pos(z1), pos(s(s(z2)))) → F(z0, pos(s(z1)), pos(s(s(z2))))
COND_F1(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(s(z1))), pos(z2))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(s(z2)))))), z0, pos(s(z1)), pos(s(s(z2))))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(s(z1), s(z2))))), z0, pos(s(s(z1))), pos(s(z2)))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, s(z2)))))), z0, pos(s(s(z1))), pos(s(z2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Instantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
COND_F1(true, z0, pos(s(z1)), pos(s(z2))) → F(z0, pos(s(s(z1))), pos(s(z2)))
COND_F1(true, z0, pos(z1), pos(s(s(z2)))) → F(z0, pos(s(z1)), pos(s(s(z2))))
COND_F1(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(s(z1))), pos(z2))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(s(z2)))))), z0, pos(s(z1)), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, s(z2)))))), z0, pos(s(s(z1))), pos(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule F(z0, pos(s(z1)), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2)) we obtained the following new rules [LPAR04]:

F(z0, pos(s(s(z1))), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(s(z1), z2)))), z0, pos(s(s(z1))), pos(z2))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
F(z0, pos(s(x1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(s(plus_nat(x1, s(s(z2)))))), z0, pos(s(x1)), pos(s(s(z2))))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(s(z1), s(z2))))), z0, pos(s(s(z1))), pos(s(z2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Instantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
COND_F1(true, z0, pos(s(z1)), pos(s(z2))) → F(z0, pos(s(s(z1))), pos(s(z2)))
COND_F1(true, z0, pos(z1), pos(s(s(z2)))) → F(z0, pos(s(z1)), pos(s(s(z2))))
COND_F1(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(s(z1))), pos(z2))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(s(z2)))))), z0, pos(s(z1)), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, s(z2)))))), z0, pos(s(s(z1))), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(z2)) → COND_F(greater_int(z0, pos(s(plus_nat(s(z1), z2)))), z0, pos(s(s(z1))), pos(z2))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(s(z1), s(z2))))), z0, pos(s(s(z1))), pos(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

F(z0, pos(s(s(z1))), pos(z2)) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, z2))))), z0, pos(s(s(z1))), pos(z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Instantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
COND_F1(true, z0, pos(s(z1)), pos(s(z2))) → F(z0, pos(s(s(z1))), pos(s(z2)))
COND_F1(true, z0, pos(z1), pos(s(s(z2)))) → F(z0, pos(s(z1)), pos(s(s(z2))))
COND_F1(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(s(z1))), pos(z2))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(s(z2)))))), z0, pos(s(z1)), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, s(z2)))))), z0, pos(s(s(z1))), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(s(z1), s(z2))))), z0, pos(s(s(z1))), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(z2)) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, z2))))), z0, pos(s(s(z1))), pos(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, s(z2)))))), z0, pos(s(s(z1))), pos(s(z2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Instantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
COND_F1(true, z0, pos(s(z1)), pos(s(z2))) → F(z0, pos(s(s(z1))), pos(s(z2)))
COND_F1(true, z0, pos(z1), pos(s(s(z2)))) → F(z0, pos(s(z1)), pos(s(s(z2))))
COND_F1(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(s(z1))), pos(z2))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(s(z2)))))), z0, pos(s(z1)), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, s(z2)))))), z0, pos(s(s(z1))), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(z2)) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, z2))))), z0, pos(s(s(z1))), pos(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule F(z0, pos(s(z1)), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, z2)))), z0, pos(s(z1)), pos(z2)) we obtained the following new rules [LPAR04]:

F(z0, pos(s(s(z1))), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(s(z1), z2)))), z0, pos(s(s(z1))), pos(z2))
F(z0, pos(s(x1)), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(s(plus_nat(x1, s(s(z2)))))), z0, pos(s(x1)), pos(s(s(z2))))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(s(z1), s(z2))))), z0, pos(s(s(z1))), pos(s(z2)))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Instantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
COND_F1(true, z0, pos(s(z1)), pos(s(z2))) → F(z0, pos(s(s(z1))), pos(s(z2)))
COND_F1(true, z0, pos(z1), pos(s(s(z2)))) → F(z0, pos(s(z1)), pos(s(s(z2))))
COND_F1(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(s(z1))), pos(z2))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(s(z2)))))), z0, pos(s(z1)), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, s(z2)))))), z0, pos(s(s(z1))), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(z2)) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, z2))))), z0, pos(s(s(z1))), pos(z2))
F(z0, pos(s(s(z1))), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(s(z1), z2)))), z0, pos(s(s(z1))), pos(z2))
F(z0, pos(s(x1)), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(s(plus_nat(x1, s(s(z2)))))), z0, pos(s(x1)), pos(s(s(z2))))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(s(z1), s(z2))))), z0, pos(s(s(z1))), pos(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule F(z0, pos(s(s(z1))), pos(z2)) → COND_F1(greater_int(z0, pos(s(plus_nat(s(z1), z2)))), z0, pos(s(s(z1))), pos(z2)) at position [0,1,0,0] we obtained the following new rules [LPAR04]:

F(z0, pos(s(s(z1))), pos(z2)) → COND_F1(greater_int(z0, pos(s(s(plus_nat(z1, z2))))), z0, pos(s(s(z1))), pos(z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Instantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
COND_F1(true, z0, pos(s(z1)), pos(s(z2))) → F(z0, pos(s(s(z1))), pos(s(z2)))
COND_F1(true, z0, pos(z1), pos(s(s(z2)))) → F(z0, pos(s(z1)), pos(s(s(z2))))
COND_F1(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(s(z1))), pos(z2))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(s(z2)))))), z0, pos(s(z1)), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, s(z2)))))), z0, pos(s(s(z1))), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(z2)) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, z2))))), z0, pos(s(s(z1))), pos(z2))
F(z0, pos(s(x1)), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(s(plus_nat(x1, s(s(z2)))))), z0, pos(s(x1)), pos(s(s(z2))))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(s(z1), s(z2))))), z0, pos(s(s(z1))), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(z2)) → COND_F1(greater_int(z0, pos(s(s(plus_nat(z1, z2))))), z0, pos(s(s(z1))), pos(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(s(z1), s(z2))))), z0, pos(s(s(z1))), pos(s(z2))) at position [0,1,0,0] we obtained the following new rules [LPAR04]:

F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(s(plus_nat(z1, s(z2)))))), z0, pos(s(s(z1))), pos(s(z2)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ Instantiation

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Instantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ Rewriting

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Rewriting

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Rewriting

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                          ↳ QDP

                                                                                          ↳ QDP

                                                                                ↳ QDP

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

COND_F(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(z1)), pos(s(z2)))
COND_F(true, z0, pos(z1), pos(s(z2))) → F(z0, pos(z1), pos(s(s(z2))))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
COND_F1(true, z0, pos(s(z1)), pos(s(z2))) → F(z0, pos(s(s(z1))), pos(s(z2)))
COND_F1(true, z0, pos(z1), pos(s(s(z2)))) → F(z0, pos(s(z1)), pos(s(s(z2))))
COND_F1(true, z0, pos(s(z1)), pos(z2)) → F(z0, pos(s(s(z1))), pos(z2))
F(z0, pos(z1), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(plus_nat(z1, s(s(z2))))), z0, pos(z1), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(s(z2)))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(s(z2)))))), z0, pos(s(z1)), pos(s(s(z2))))
F(z0, pos(s(z1)), pos(s(z2))) → COND_F(greater_int(z0, pos(s(plus_nat(z1, s(z2))))), z0, pos(s(z1)), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, s(z2)))))), z0, pos(s(s(z1))), pos(s(z2)))
F(z0, pos(s(s(z1))), pos(z2)) → COND_F(greater_int(z0, pos(s(s(plus_nat(z1, z2))))), z0, pos(s(s(z1))), pos(z2))
F(z0, pos(s(x1)), pos(s(s(z2)))) → COND_F1(greater_int(z0, pos(s(plus_nat(x1, s(s(z2)))))), z0, pos(s(x1)), pos(s(s(z2))))
F(z0, pos(s(s(z1))), pos(z2)) → COND_F1(greater_int(z0, pos(s(s(plus_nat(z1, z2))))), z0, pos(s(s(z1))), pos(z2))
F(z0, pos(s(s(z1))), pos(s(z2))) → COND_F1(greater_int(z0, pos(s(s(plus_nat(z1, s(z2)))))), z0, pos(s(s(z1))), pos(s(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                ↳ QDP

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

F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
COND_F(true, z0, pos(z1), neg(z2)) → F(z0, pos(z1), minus_nat(s(0), z2))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
COND_F1(true, z0, pos(z1), neg(z2)) → F(z0, pos(s(z1)), neg(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                ↳ QDP

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

F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
COND_F(true, z0, pos(z1), neg(z2)) → F(z0, pos(z1), minus_nat(s(0), z2))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
COND_F1(true, z0, pos(z1), neg(z2)) → F(z0, pos(s(z1)), neg(z2))

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

The set Q consists of the following terms:

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

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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                ↳ QDP

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

F(y0, pos(x0), neg(x1)) → COND_F(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
COND_F(true, z0, pos(z1), neg(z2)) → F(z0, pos(z1), minus_nat(s(0), z2))
F(y0, pos(x0), neg(x1)) → COND_F1(greater_int(y0, minus_nat(x0, x1)), y0, pos(x0), neg(x1))
COND_F1(true, z0, pos(z1), neg(z2)) → F(z0, pos(s(z1)), neg(z2))

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)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

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

COND_F(true, z0, neg(z1), neg(z2)) → F(z0, neg(z1), minus_nat(s(0), z2))
F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
COND_F(true, z0, neg(z1), pos(z2)) → F(z0, neg(z1), pos(s(z2)))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
COND_F1(true, y0, neg(x1), y2) → F(y0, minus_nat(s(0), x1), y2)
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_F1(true, y0, neg(x1), y2) → F(y0, minus_nat(s(0), x1), y2) we obtained the following new rules [LPAR04]:

COND_F1(true, z0, neg(z1), pos(z2)) → F(z0, minus_nat(s(0), z1), pos(z2))
COND_F1(true, z0, neg(z1), neg(z2)) → F(z0, minus_nat(s(0), z1), neg(z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

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

COND_F(true, z0, neg(z1), neg(z2)) → F(z0, neg(z1), minus_nat(s(0), z2))
F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
COND_F(true, z0, neg(z1), pos(z2)) → F(z0, neg(z1), pos(s(z2)))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
COND_F1(true, z0, neg(z1), pos(z2)) → F(z0, minus_nat(s(0), z1), pos(z2))
COND_F1(true, z0, neg(z1), neg(z2)) → F(z0, minus_nat(s(0), z1), neg(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                          ↳ QDP

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

COND_F(true, z0, neg(z1), pos(z2)) → F(z0, neg(z1), pos(s(z2)))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
COND_F1(true, z0, neg(z1), pos(z2)) → F(z0, minus_nat(s(0), z1), pos(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                          ↳ QDP

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

COND_F(true, z0, neg(z1), pos(z2)) → F(z0, neg(z1), pos(s(z2)))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
COND_F1(true, z0, neg(z1), pos(z2)) → F(z0, minus_nat(s(0), z1), pos(z2))

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

The set Q consists of the following terms:

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

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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                          ↳ QDP

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

COND_F(true, z0, neg(z1), pos(z2)) → F(z0, neg(z1), pos(s(z2)))
F(y0, neg(x0), pos(x1)) → COND_F(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
F(y0, neg(x0), pos(x1)) → COND_F1(greater_int(y0, minus_nat(x1, x0)), y0, neg(x0), pos(x1))
COND_F1(true, z0, neg(z1), pos(z2)) → F(z0, minus_nat(s(0), z1), pos(z2))

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)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

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

F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
COND_F(true, z0, neg(z1), neg(z2)) → F(z0, neg(z1), minus_nat(s(0), z2))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
COND_F1(true, z0, neg(z1), neg(z2)) → F(z0, minus_nat(s(0), z1), neg(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                ↳ QDP

                                                                                  ↳ Instantiation

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ AND

                                                                                          ↳ QDP

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

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

F(y0, neg(x0), neg(x1)) → COND_F(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
COND_F(true, z0, neg(z1), neg(z2)) → F(z0, neg(z1), minus_nat(s(0), z2))
F(y0, neg(x0), neg(x1)) → COND_F1(greater_int(y0, neg(plus_nat(x0, x1))), y0, neg(x0), neg(x1))
COND_F1(true, z0, neg(z1), neg(z2)) → F(z0, minus_nat(s(0), z1), neg(z2))

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

The set Q consists of the following terms:

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

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