Termination proof of /tmp/tmpbnsYHA/02.itrs

Termination of the given ITRSProblem could successfully be proven:

↳ ITRS

  ↳ ITRStoQTRSProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

Represented integers and predefined function symbols by Terms

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

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

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

The set Q consists of the following terms:

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

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

EVAL(x, y) → COND_EVAL(greater_int(x, y), x, y)
EVAL(x, y) → GREATER_INT(x, y)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), plus_int(pos(s(0)), y))
COND_EVAL(true, x, y) → MINUS_INT(x, pos(s(0)))
COND_EVAL(true, x, y) → PLUS_INT(pos(s(0)), 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))
MINUS_INT(pos(x), pos(y)) → MINUS_NAT(x, y)
MINUS_INT(neg(x), neg(y)) → MINUS_NAT(y, x)
MINUS_INT(neg(x), pos(y)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
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)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

EVAL(x, y) → COND_EVAL(greater_int(x, y), x, y)
EVAL(x, y) → GREATER_INT(x, y)
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), plus_int(pos(s(0)), y))
COND_EVAL(true, x, y) → MINUS_INT(x, pos(s(0)))
COND_EVAL(true, x, y) → PLUS_INT(pos(s(0)), 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))
MINUS_INT(pos(x), pos(y)) → MINUS_NAT(x, y)
MINUS_INT(neg(x), neg(y)) → MINUS_NAT(y, x)
MINUS_INT(neg(x), pos(y)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
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)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 11 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:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

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

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

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

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

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

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

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

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

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

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

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

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

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

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

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

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

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

The following dependency pairs can be deleted:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

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

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

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

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

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

↳ 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:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

[0,1]
[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:

COND_EVAL(true, x, y, x_removed) → EVAL(minus_int(x, x_removed), plus_int(x_removed, y), x_removed)
EVAL(x, y, x_removed) → COND_EVAL(greater_int(x, y), x, y, x_removed)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

[0,1]
[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:

COND_EVAL(true, x, y, x_removed) → EVAL(minus_int(x, x_removed), plus_int(x_removed, y), x_removed)
EVAL(x, y, x_removed) → COND_EVAL(greater_int(x, y), x, y, x_removed)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

EVAL(neg(s(x0)), neg(0)) → COND_EVAL(false, neg(s(x0)), neg(0))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), pos(s(x1))) → COND_EVAL(false, neg(s(x0)), pos(s(x1)))
EVAL(neg(0), neg(s(x0))) → COND_EVAL(true, neg(0), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
EVAL(neg(0), pos(0)) → COND_EVAL(false, neg(0), pos(0))
EVAL(pos(0), neg(0)) → COND_EVAL(false, pos(0), neg(0))
EVAL(neg(0), neg(0)) → COND_EVAL(false, neg(0), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(0), pos(0)) → COND_EVAL(false, pos(0), pos(0))
EVAL(neg(0), pos(s(x0))) → COND_EVAL(false, neg(0), pos(s(x0)))
EVAL(neg(s(x0)), pos(0)) → COND_EVAL(false, neg(s(x0)), pos(0))
EVAL(pos(0), pos(s(x0))) → COND_EVAL(false, pos(0), pos(s(x0)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL(true, pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL(true, pos(s(x0)), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

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

COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), plus_int(pos(s(0)), y))
EVAL(neg(s(x0)), neg(0)) → COND_EVAL(false, neg(s(x0)), neg(0))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(neg(s(x0)), pos(s(x1))) → COND_EVAL(false, neg(s(x0)), pos(s(x1)))
EVAL(neg(0), neg(s(x0))) → COND_EVAL(true, neg(0), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
EVAL(neg(0), pos(0)) → COND_EVAL(false, neg(0), pos(0))
EVAL(pos(0), neg(0)) → COND_EVAL(false, pos(0), neg(0))
EVAL(neg(0), neg(0)) → COND_EVAL(false, neg(0), neg(0))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(0), pos(0)) → COND_EVAL(false, pos(0), pos(0))
EVAL(neg(0), pos(s(x0))) → COND_EVAL(false, neg(0), pos(s(x0)))
EVAL(neg(s(x0)), pos(0)) → COND_EVAL(false, neg(s(x0)), pos(0))
EVAL(pos(0), pos(s(x0))) → COND_EVAL(false, pos(0), pos(s(x0)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL(true, pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL(true, pos(s(x0)), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

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

EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), plus_int(pos(s(0)), y))
EVAL(neg(0), neg(s(x0))) → COND_EVAL(true, neg(0), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL(true, pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL(true, pos(s(x0)), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

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

EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND_EVAL(true, x, y) → EVAL(minus_int(x, pos(s(0))), plus_int(pos(s(0)), y))
EVAL(neg(0), neg(s(x0))) → COND_EVAL(true, neg(0), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL(true, pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL(true, pos(s(x0)), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), plus_int(pos(s(0)), y1))
COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), plus_int(pos(s(0)), y1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

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

EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(neg(0), neg(s(x0))) → COND_EVAL(true, neg(0), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL(true, pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL(true, pos(s(x0)), neg(0))
COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), plus_int(pos(s(0)), y1))
COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), plus_int(pos(s(0)), y1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                            ↳ QDP

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

COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), plus_int(pos(s(0)), y1))
EVAL(neg(0), neg(s(x0))) → COND_EVAL(true, neg(0), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                            ↳ QDP

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

COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), plus_int(pos(s(0)), y1))
EVAL(neg(0), neg(s(x0))) → COND_EVAL(true, neg(0), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                            ↳ QDP

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

COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), plus_int(pos(s(0)), y1))
EVAL(neg(0), neg(s(x0))) → COND_EVAL(true, neg(0), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL(true, neg(x0), y1) → EVAL(neg(plus_nat(x0, s(0))), plus_int(pos(s(0)), y1)) at position [0,0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(0), y1) → EVAL(neg(s(0)), plus_int(pos(s(0)), y1))
COND_EVAL(true, neg(s(x0)), y1) → EVAL(neg(s(plus_nat(x0, s(0)))), plus_int(pos(s(0)), y1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                            ↳ QDP

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

EVAL(neg(0), neg(s(x0))) → COND_EVAL(true, neg(0), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND_EVAL(true, neg(0), y1) → EVAL(neg(s(0)), plus_int(pos(s(0)), y1))
COND_EVAL(true, neg(s(x0)), y1) → EVAL(neg(s(plus_nat(x0, s(0)))), plus_int(pos(s(0)), y1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                            ↳ QDP

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

COND_EVAL(true, neg(s(x0)), y1) → EVAL(neg(s(plus_nat(x0, s(0)))), plus_int(pos(s(0)), y1))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL(true, neg(s(x0)), y1) → EVAL(neg(s(plus_nat(x0, s(0)))), plus_int(pos(s(0)), y1)) we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(plus_nat(z0, s(0)))), plus_int(pos(s(0)), neg(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                            ↳ QDP

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

EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(plus_nat(z0, s(0)))), plus_int(pos(s(0)), neg(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                            ↳ QDP

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

EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(plus_nat(z0, s(0)))), plus_int(pos(s(0)), neg(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(plus_nat(z0, s(0)))), minus_nat(s(0), s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ UsableRulesProof

                                            ↳ QDP

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

EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(plus_nat(z0, s(0)))), minus_nat(s(0), s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ UsableRulesProof

                                                                            ↳ QDP

                                                                              ↳ QReductionProof

                                            ↳ QDP

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

EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(plus_nat(z0, s(0)))), minus_nat(s(0), s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ UsableRulesProof

                                                                            ↳ QDP

                                                                              ↳ QReductionProof

                                                                                ↳ QDP

                                                                                  ↳ Rewriting

                                            ↳ QDP

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

EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(plus_nat(z0, s(0)))), minus_nat(s(0), s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(plus_nat(z0, s(0)))), minus_nat(s(0), s(z1))) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(plus_nat(z0, s(0)))), minus_nat(0, z1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ UsableRulesProof

                                                                            ↳ QDP

                                                                              ↳ QReductionProof

                                                                                ↳ QDP

                                                                                  ↳ Rewriting

                                                                                    ↳ QDP

                                                                                      ↳ UsableRulesProof

                                            ↳ QDP

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

EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(plus_nat(z0, s(0)))), minus_nat(0, z1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ UsableRulesProof

                                                                            ↳ QDP

                                                                              ↳ QReductionProof

                                                                                ↳ QDP

                                                                                  ↳ Rewriting

                                                                                    ↳ QDP

                                                                                      ↳ UsableRulesProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                            ↳ QDP

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

EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(plus_nat(z0, s(0)))), minus_nat(0, z1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL(true, neg(s(z0)), neg(s(z1))) → EVAL(neg(s(plus_nat(z0, s(0)))), minus_nat(0, z1)) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL(true, neg(s(y0)), neg(s(s(x0)))) → EVAL(neg(s(plus_nat(y0, s(0)))), neg(s(x0)))
COND_EVAL(true, neg(s(y0)), neg(s(0))) → EVAL(neg(s(plus_nat(y0, s(0)))), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ UsableRulesProof

                                                                            ↳ QDP

                                                                              ↳ QReductionProof

                                                                                ↳ QDP

                                                                                  ↳ Rewriting

                                                                                    ↳ QDP

                                                                                      ↳ UsableRulesProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                            ↳ QDP

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

EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND_EVAL(true, neg(s(y0)), neg(s(s(x0)))) → EVAL(neg(s(plus_nat(y0, s(0)))), neg(s(x0)))
COND_EVAL(true, neg(s(y0)), neg(s(0))) → EVAL(neg(s(plus_nat(y0, s(0)))), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ UsableRulesProof

                                                                            ↳ QDP

                                                                              ↳ QReductionProof

                                                                                ↳ QDP

                                                                                  ↳ Rewriting

                                                                                    ↳ QDP

                                                                                      ↳ UsableRulesProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                            ↳ QDP

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

COND_EVAL(true, neg(s(y0)), neg(s(s(x0)))) → EVAL(neg(s(plus_nat(y0, s(0)))), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(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))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ UsableRulesProof

                                                                            ↳ QDP

                                                                              ↳ QReductionProof

                                                                                ↳ QDP

                                                                                  ↳ Rewriting

                                                                                    ↳ QDP

                                                                                      ↳ UsableRulesProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ QReductionProof

                                            ↳ QDP

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

COND_EVAL(true, neg(s(y0)), neg(s(s(x0)))) → EVAL(neg(s(plus_nat(y0, s(0)))), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(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(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

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

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ UsableRulesProof

                                                                            ↳ QDP

                                                                              ↳ QReductionProof

                                                                                ↳ QDP

                                                                                  ↳ Rewriting

                                                                                    ↳ QDP

                                                                                      ↳ UsableRulesProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ QReductionProof

                                                                                                        ↳ QDP

                                                                                                          ↳ ForwardInstantiation

                                            ↳ QDP

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

COND_EVAL(true, neg(s(y0)), neg(s(s(x0)))) → EVAL(neg(s(plus_nat(y0, s(0)))), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(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(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule EVAL(neg(s(x0)), neg(s(x1))) → COND_EVAL(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))) we obtained the following new rules [LPAR04]:

EVAL(neg(s(x0)), neg(s(s(y_2)))) → COND_EVAL(greater_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ UsableRulesProof

                                                                            ↳ QDP

                                                                              ↳ QReductionProof

                                                                                ↳ QDP

                                                                                  ↳ Rewriting

                                                                                    ↳ QDP

                                                                                      ↳ UsableRulesProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ QReductionProof

                                                                                                        ↳ QDP

                                                                                                          ↳ ForwardInstantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ QDPOrderProof

                                            ↳ QDP

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

COND_EVAL(true, neg(s(y0)), neg(s(s(x0)))) → EVAL(neg(s(plus_nat(y0, s(0)))), neg(s(x0)))
EVAL(neg(s(x0)), neg(s(s(y_2)))) → COND_EVAL(greater_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))

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

The set Q consists of the following terms:

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

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

The following pairs can be oriented strictly and are deleted.

COND_EVAL(true, neg(s(y0)), neg(s(s(x0)))) → EVAL(neg(s(plus_nat(y0, s(0)))), neg(s(x0)))

The remaining pairs can at least be oriented weakly.

EVAL(neg(s(x0)), neg(s(s(y_2)))) → COND_EVAL(greater_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND_EVAL(x₁, x₂, x₃)) = x₃
POL(EVAL(x₁, x₂)) = x₂
POL(false) = 0
POL(greater_int(x₁, x₂)) = 0
POL(neg(x₁)) = x₁
POL(plus_nat(x₁, x₂)) = 0
POL(s(x₁)) = 1 + x₁
POL(true) = 0

The following usable rules [FROCOS05] were oriented: none

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ UsableRulesProof

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ UsableRulesProof

                                                                            ↳ QDP

                                                                              ↳ QReductionProof

                                                                                ↳ QDP

                                                                                  ↳ Rewriting

                                                                                    ↳ QDP

                                                                                      ↳ UsableRulesProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ QReductionProof

                                                                                                        ↳ QDP

                                                                                                          ↳ ForwardInstantiation

                                                                                                            ↳ QDP

                                                                                                              ↳ QDPOrderProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                            ↳ QDP

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

EVAL(neg(s(x0)), neg(s(s(y_2)))) → COND_EVAL(greater_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

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

COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), plus_int(pos(s(0)), y1))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL(true, pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL(true, pos(s(x0)), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

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

COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), plus_int(pos(s(0)), y1))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL(true, pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL(true, pos(s(x0)), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

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

COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), plus_int(pos(s(0)), y1))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL(true, pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL(true, pos(s(x0)), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL(true, pos(x0), y1) → EVAL(minus_nat(x0, s(0)), plus_int(pos(s(0)), y1)) at position [0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(x0)), y1) → EVAL(minus_nat(x0, 0), plus_int(pos(s(0)), y1))
COND_EVAL(true, pos(0), y1) → EVAL(neg(s(0)), plus_int(pos(s(0)), y1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

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

EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(0), neg(s(x0))) → COND_EVAL(true, pos(0), neg(s(x0)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL(true, pos(s(x0)), neg(0))
COND_EVAL(true, pos(s(x0)), y1) → EVAL(minus_nat(x0, 0), plus_int(pos(s(0)), y1))
COND_EVAL(true, pos(0), y1) → EVAL(neg(s(0)), plus_int(pos(s(0)), y1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

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

COND_EVAL(true, pos(s(x0)), y1) → EVAL(minus_nat(x0, 0), plus_int(pos(s(0)), y1))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL(true, pos(s(x0)), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL(true, pos(s(x0)), y1) → EVAL(minus_nat(x0, 0), plus_int(pos(s(0)), y1)) we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), pos(0)))
COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), pos(s(z1))))
COND_EVAL(true, pos(s(z0)), neg(0)) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), neg(0)))
COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), neg(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

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

EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL(true, pos(s(x0)), neg(0))
COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), pos(0)))
COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), pos(s(z1))))
COND_EVAL(true, pos(s(z0)), neg(0)) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), neg(0)))
COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), neg(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), pos(0))) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), 0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

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

EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL(true, pos(s(x0)), neg(0))
COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), pos(s(z1))))
COND_EVAL(true, pos(s(z0)), neg(0)) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), neg(0)))
COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), neg(s(z1))))
COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), 0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), pos(s(z1)))) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

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

EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
EVAL(pos(s(x0)), neg(0)) → COND_EVAL(true, pos(s(x0)), neg(0))
COND_EVAL(true, pos(s(z0)), neg(0)) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), neg(0)))
COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), neg(s(z1))))
COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), 0)))
COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                              ↳ QDP

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

COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), s(z1))))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), 0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                              ↳ QDP

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

COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), s(z1))))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), 0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                              ↳ QDP

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

COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), s(z1))))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), 0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(plus_nat(0, s(z1)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

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

EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), 0)))
COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(plus_nat(0, s(z1)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(plus_nat(0, s(z1)))))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(plus_nat(0, s(z1)))))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL(true, pos(s(z0)), pos(s(z1))) → EVAL(minus_nat(z0, 0), pos(s(s(z1)))) at position [0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(s(y1))))
COND_EVAL(true, pos(s(0)), pos(s(y1))) → EVAL(pos(0), pos(s(s(y1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(s(y1))))
COND_EVAL(true, pos(s(0)), pos(s(y1))) → EVAL(pos(0), pos(s(s(y1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ UsableRulesProof

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(s(y1))))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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)))
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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ UsableRulesProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ QReductionProof

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(s(y1))))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ UsableRulesProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ QReductionProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(s(y1))))
EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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 EVAL(pos(s(x0)), pos(s(x1))) → COND_EVAL(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))) we obtained the following new rules [LPAR04]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ UsableRulesProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ QReductionProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Instantiation

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

COND_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(s(y1))))
EVAL(pos(s(z0)), pos(s(s(z1)))) → COND_EVAL(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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_EVAL(true, pos(s(s(x0))), pos(s(y1))) → EVAL(pos(s(x0)), pos(s(s(y1)))) we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(s(x0))), pos(s(s(z1)))) → EVAL(pos(s(x0)), pos(s(s(s(z1)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ UsableRulesProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ QReductionProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Instantiation

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

EVAL(pos(s(z0)), pos(s(s(z1)))) → COND_EVAL(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
COND_EVAL(true, pos(s(s(x0))), pos(s(s(z1)))) → EVAL(pos(s(x0)), pos(s(s(s(z1)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ UsableRulesProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ QReductionProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Instantiation

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ ForwardInstantiation

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

COND_EVAL(true, pos(s(s(x0))), pos(s(s(z1)))) → EVAL(pos(s(x0)), pos(s(s(s(z1)))))
EVAL(pos(s(z0)), pos(s(s(s(z1))))) → COND_EVAL(greater_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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 forward instantiating [JAR06] the rule EVAL(pos(s(z0)), pos(s(s(s(z1))))) → COND_EVAL(greater_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1))))) we obtained the following new rules [LPAR04]:

EVAL(pos(s(s(y_1))), pos(s(s(s(x1))))) → COND_EVAL(greater_int(pos(s(y_1)), pos(s(s(x1)))), pos(s(s(y_1))), pos(s(s(s(x1)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ UsableRulesProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ QReductionProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Instantiation

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ Rewriting

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

COND_EVAL(true, pos(s(s(x0))), pos(s(s(z1)))) → EVAL(pos(s(x0)), pos(s(s(s(z1)))))
EVAL(pos(s(s(y_1))), pos(s(s(s(x1))))) → COND_EVAL(greater_int(pos(s(y_1)), pos(s(s(x1)))), pos(s(s(y_1))), pos(s(s(s(x1)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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 EVAL(pos(s(s(y_1))), pos(s(s(s(x1))))) → COND_EVAL(greater_int(pos(s(y_1)), pos(s(s(x1)))), pos(s(s(y_1))), pos(s(s(s(x1))))) at position [0] we obtained the following new rules [LPAR04]:

EVAL(pos(s(s(y_1))), pos(s(s(s(x1))))) → COND_EVAL(greater_int(pos(y_1), pos(s(x1))), pos(s(s(y_1))), pos(s(s(s(x1)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ UsableRulesProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ QReductionProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Instantiation

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ Rewriting

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

COND_EVAL(true, pos(s(s(x0))), pos(s(s(z1)))) → EVAL(pos(s(x0)), pos(s(s(s(z1)))))
EVAL(pos(s(s(y_1))), pos(s(s(s(x1))))) → COND_EVAL(greater_int(pos(y_1), pos(s(x1))), pos(s(s(y_1))), pos(s(s(s(x1)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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 forward instantiating [JAR06] the rule COND_EVAL(true, pos(s(s(x0))), pos(s(s(z1)))) → EVAL(pos(s(x0)), pos(s(s(s(z1))))) we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(s(s(y_0)))), pos(s(s(x1)))) → EVAL(pos(s(s(y_0))), pos(s(s(s(x1)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ UsableRulesProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ QReductionProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Instantiation

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ Rewriting

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

EVAL(pos(s(s(y_1))), pos(s(s(s(x1))))) → COND_EVAL(greater_int(pos(y_1), pos(s(x1))), pos(s(s(y_1))), pos(s(s(s(x1)))))
COND_EVAL(true, pos(s(s(s(y_0)))), pos(s(s(x1)))) → EVAL(pos(s(s(y_0))), pos(s(s(s(x1)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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 use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

COND_EVAL(true, pos(s(s(s(y_0)))), pos(s(s(x1)))) → EVAL(pos(s(s(y_0))), pos(s(s(s(x1)))))

The remaining pairs can at least be oriented weakly.

EVAL(pos(s(s(y_1))), pos(s(s(s(x1))))) → COND_EVAL(greater_int(pos(y_1), pos(s(x1))), pos(s(s(y_1))), pos(s(s(s(x1)))))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND_EVAL(x₁, x₂, x₃)) = x₂
POL(EVAL(x₁, x₂)) = x₁
POL(false) = 0
POL(greater_int(x₁, x₂)) = 0
POL(pos(x₁)) = 1 + x₁
POL(s(x₁)) = 1 + x₁
POL(true) = 0

The following usable rules [FROCOS05] were oriented: none

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ QReductionProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ UsableRulesProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ QReductionProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Instantiation

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ Rewriting

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ ForwardInstantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                              ↳ QDP

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

EVAL(pos(s(s(y_1))), pos(s(s(s(x1))))) → COND_EVAL(greater_int(pos(y_1), pos(s(x1))), pos(s(s(y_1))), pos(s(s(s(x1)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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 0 SCCs with 1 less node.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                              ↳ QDP

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

EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), 0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ QReductionProof

                                                                              ↳ QDP

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

EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), 0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ QReductionProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Rewriting

                                                                              ↳ QDP

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

EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), pos(plus_nat(s(0), 0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), pos(s(plus_nat(0, 0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ AND

                                                                                                ↳ QDP

                                                                                                ↳ QDP

                                                                                                  ↳ UsableRulesProof

                                                                                                    ↳ QDP

                                                                                                      ↳ QReductionProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Rewriting

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                              ↳ QDP

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

EVAL(pos(s(x0)), pos(0)) → COND_EVAL(true, pos(s(x0)), pos(0))
COND_EVAL(true, pos(s(z0)), pos(0)) → EVAL(minus_nat(z0, 0), pos(s(plus_nat(0, 0))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

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

EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), neg(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

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

EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), neg(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

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

EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), neg(s(z1))))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, 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))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), plus_int(pos(s(0)), neg(s(z1)))) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), minus_nat(s(0), s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

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

EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), minus_nat(s(0), s(z1)))

The TRS R consists of the following rules:

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

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

EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), minus_nat(s(0), s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

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

EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), minus_nat(s(0), s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), minus_nat(s(0), s(z1))) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), minus_nat(0, z1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

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

EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), minus_nat(0, z1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

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

EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), minus_nat(0, z1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL(true, pos(s(z0)), neg(s(z1))) → EVAL(minus_nat(z0, 0), minus_nat(0, z1)) at position [0] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(s(x0))), neg(s(y1))) → EVAL(pos(s(x0)), minus_nat(0, y1))
COND_EVAL(true, pos(s(0)), neg(s(y1))) → EVAL(pos(0), minus_nat(0, y1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

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

EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
COND_EVAL(true, pos(s(s(x0))), neg(s(y1))) → EVAL(pos(s(x0)), minus_nat(0, y1))
COND_EVAL(true, pos(s(0)), neg(s(y1))) → EVAL(pos(0), minus_nat(0, y1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

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

COND_EVAL(true, pos(s(s(x0))), neg(s(y1))) → EVAL(pos(s(x0)), minus_nat(0, y1))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

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

COND_EVAL(true, pos(s(s(x0))), neg(s(y1))) → EVAL(pos(s(x0)), minus_nat(0, y1))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL(true, pos(s(s(x0))), neg(s(y1))) → EVAL(pos(s(x0)), minus_nat(0, y1)) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(s(y0))), neg(s(s(x0)))) → EVAL(pos(s(y0)), neg(s(x0)))
COND_EVAL(true, pos(s(s(y0))), neg(s(0))) → EVAL(pos(s(y0)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

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

EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))
COND_EVAL(true, pos(s(s(y0))), neg(s(s(x0)))) → EVAL(pos(s(y0)), neg(s(x0)))
COND_EVAL(true, pos(s(s(y0))), neg(s(0))) → EVAL(pos(s(y0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

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

COND_EVAL(true, pos(s(s(y0))), neg(s(s(x0)))) → EVAL(pos(s(y0)), neg(s(x0)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

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

COND_EVAL(true, pos(s(s(y0))), neg(s(s(x0)))) → EVAL(pos(s(y0)), neg(s(x0)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))

R is empty.
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))

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

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ ForwardInstantiation

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

COND_EVAL(true, pos(s(s(y0))), neg(s(s(x0)))) → EVAL(pos(s(y0)), neg(s(x0)))
EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1)))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule EVAL(pos(s(x0)), neg(s(x1))) → COND_EVAL(true, pos(s(x0)), neg(s(x1))) we obtained the following new rules [LPAR04]:

EVAL(pos(s(s(y_0))), neg(s(s(y_1)))) → COND_EVAL(true, pos(s(s(y_0))), neg(s(s(y_1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ ForwardInstantiation

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

COND_EVAL(true, pos(s(s(y0))), neg(s(s(x0)))) → EVAL(pos(s(y0)), neg(s(x0)))
EVAL(pos(s(s(y_0))), neg(s(s(y_1)))) → COND_EVAL(true, pos(s(s(y_0))), neg(s(s(y_1))))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND_EVAL(true, pos(s(s(y0))), neg(s(s(x0)))) → EVAL(pos(s(y0)), neg(s(x0))) we obtained the following new rules [LPAR04]:

COND_EVAL(true, pos(s(s(s(y_0)))), neg(s(s(s(y_1))))) → EVAL(pos(s(s(y_0))), neg(s(s(y_1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ MRRProof

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

EVAL(pos(s(s(y_0))), neg(s(s(y_1)))) → COND_EVAL(true, pos(s(s(y_0))), neg(s(s(y_1))))
COND_EVAL(true, pos(s(s(s(y_0)))), neg(s(s(s(y_1))))) → EVAL(pos(s(s(y_0))), neg(s(s(y_1))))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

EVAL(pos(s(s(y_0))), neg(s(s(y_1)))) → COND_EVAL(true, pos(s(s(y_0))), neg(s(s(y_1))))
COND_EVAL(true, pos(s(s(s(y_0)))), neg(s(s(s(y_1))))) → EVAL(pos(s(s(y_0))), neg(s(s(y_1))))

Used ordering: Polynomial interpretation [POLO]:

POL(COND_EVAL(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(EVAL(x₁, x₂)) = 2·x₁ + 2·x₂
POL(neg(x₁)) = 2·x₁
POL(pos(x₁)) = 2·x₁
POL(s(x₁)) = 2 + 2·x₁
POL(true) = 0

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ AND

                                            ↳ QDP

                                            ↳ QDP

                                              ↳ UsableRulesProof

                                                ↳ QDP

                                                  ↳ QReductionProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Instantiation

                                                                ↳ QDP

                                                                  ↳ Rewriting

                                                                    ↳ QDP

                                                                      ↳ Rewriting

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ AND

                                                                              ↳ QDP

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ UsableRulesProof

                                                                                              ↳ QDP

                                                                                                ↳ QReductionProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ UsableRulesProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ UsableRulesProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ MRRProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ PisEmptyProof

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.