Termination proof of /tmp/tmpCfdbhW/A04.itrs

Termination of the given ITRSProblem could not be shown:

↳ ITRS

  ↳ ITRStoQTRSProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

min(u, v) → if(<@z(u, v), u, v)
cond(y, x, y) → +@z(1@z, minus(x, +@z(y, 1@z)))
if(FALSE, u, v) → v
if(TRUE, u, v) → u
minus(x, x) → 0@z
minus(x, y) → cond(min(x, y), x, y)

The set Q consists of the following terms:

min(x0, x1)
cond(x0, x1, x0)
if(FALSE, x0, x1)
if(TRUE, x0, x1)
minus(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:

min(u, v) → if(less_int(u, v), u, v)
cond(y, x, y) → plus_int(pos(s(0)), minus(x, plus_int(pos(s(0)), y)))
if(false, u, v) → v
if(true, u, v) → u
minus(x, x) → pos(0)
minus(x, y) → cond(min(x, y), x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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))

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

MIN(u, v) → IF(less_int(u, v), u, v)
MIN(u, v) → LESS_INT(u, v)
COND(y, x, y) → PLUS_INT(pos(s(0)), minus(x, plus_int(pos(s(0)), y)))
COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
COND(y, x, y) → PLUS_INT(pos(s(0)), y)
MINUS(x, y) → COND(min(x, y), x, y)
MINUS(x, y) → MIN(x, y)
LESS_INT(pos(s(x)), pos(s(y))) → LESS_INT(pos(x), pos(y))
LESS_INT(neg(s(x)), neg(s(y))) → LESS_INT(neg(x), neg(y))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)

The TRS R consists of the following rules:

min(u, v) → if(less_int(u, v), u, v)
cond(y, x, y) → plus_int(pos(s(0)), minus(x, plus_int(pos(s(0)), y)))
if(false, u, v) → v
if(true, u, v) → u
minus(x, x) → pos(0)
minus(x, y) → cond(min(x, y), x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

MIN(u, v) → IF(less_int(u, v), u, v)
MIN(u, v) → LESS_INT(u, v)
COND(y, x, y) → PLUS_INT(pos(s(0)), minus(x, plus_int(pos(s(0)), y)))
COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
COND(y, x, y) → PLUS_INT(pos(s(0)), y)
MINUS(x, y) → COND(min(x, y), x, y)
MINUS(x, y) → MIN(x, y)
LESS_INT(pos(s(x)), pos(s(y))) → LESS_INT(pos(x), pos(y))
LESS_INT(neg(s(x)), neg(s(y))) → LESS_INT(neg(x), neg(y))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)

The TRS R consists of the following rules:

min(u, v) → if(less_int(u, v), u, v)
cond(y, x, y) → plus_int(pos(s(0)), minus(x, plus_int(pos(s(0)), y)))
if(false, u, v) → v
if(true, u, v) → u
minus(x, x) → pos(0)
minus(x, y) → cond(min(x, y), x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 5 SCCs with 9 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:

min(u, v) → if(less_int(u, v), u, v)
cond(y, x, y) → plus_int(pos(s(0)), minus(x, plus_int(pos(s(0)), y)))
if(false, u, v) → v
if(true, u, v) → u
minus(x, x) → pos(0)
minus(x, y) → cond(min(x, y), x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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.
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:

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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].

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                ↳ 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:

min(u, v) → if(less_int(u, v), u, v)
cond(y, x, y) → plus_int(pos(s(0)), minus(x, plus_int(pos(s(0)), y)))
if(false, u, v) → v
if(true, u, v) → u
minus(x, x) → pos(0)
minus(x, y) → cond(min(x, y), x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                ↳ 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:

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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].

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                ↳ 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:

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

The TRS R consists of the following rules:

min(u, v) → if(less_int(u, v), u, v)
cond(y, x, y) → plus_int(pos(s(0)), minus(x, plus_int(pos(s(0)), y)))
if(false, u, v) → v
if(true, u, v) → u
minus(x, x) → pos(0)
minus(x, y) → cond(min(x, y), x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

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

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

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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].

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

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

LESS_INT(neg(s(x)), neg(s(y))) → LESS_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:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(LESS_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:

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

The TRS R consists of the following rules:

min(u, v) → if(less_int(u, v), u, v)
cond(y, x, y) → plus_int(pos(s(0)), minus(x, plus_int(pos(s(0)), y)))
if(false, u, v) → v
if(true, u, v) → u
minus(x, x) → pos(0)
minus(x, y) → cond(min(x, y), x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

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

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

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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].

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

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

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(LESS_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

                ↳ UsableRulesProof

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

MINUS(x, y) → COND(min(x, y), x, y)
COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))

The TRS R consists of the following rules:

min(u, v) → if(less_int(u, v), u, v)
cond(y, x, y) → plus_int(pos(s(0)), minus(x, plus_int(pos(s(0)), y)))
if(false, u, v) → v
if(true, u, v) → u
minus(x, x) → pos(0)
minus(x, y) → cond(min(x, y), x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                ↳ UsableRulesProof

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

MINUS(x, y) → COND(min(x, y), x, y)
COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))

The TRS R consists of the following rules:

min(u, v) → if(less_int(u, v), u, v)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u
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:

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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].

cond(x0, x1, x0)
minus(x0, x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                ↳ UsableRulesProof

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

MINUS(x, y) → COND(min(x, y), x, y)
COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))

The TRS R consists of the following rules:

min(u, v) → if(less_int(u, v), u, v)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u
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:

min(x0, x1)
if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(x, y) → COND(min(x, y), x, y) at position [0] we obtained the following new rules [LPAR04]:

MINUS(x, y) → COND(if(less_int(x, y), x, y), x, y)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(x, y) → COND(if(less_int(x, y), x, y), x, y)

The TRS R consists of the following rules:

min(u, v) → if(less_int(u, v), u, v)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u
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:

min(x0, x1)
if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(x, y) → COND(if(less_int(x, y), x, y), x, y)

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

min(x0, x1)
if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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].

min(x0, x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(x, y) → COND(if(less_int(x, y), x, y), x, y)

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(x, y) → COND(if(less_int(x, y), x, y), x, y) at position [] we obtained the following new rules [LPAR04]:

MINUS(neg(0), pos(s(x0))) → COND(if(true, neg(0), pos(s(x0))), neg(0), pos(s(x0)))
MINUS(neg(s(x0)), pos(0)) → COND(if(true, neg(s(x0)), pos(0)), neg(s(x0)), pos(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(if(false, pos(s(x0)), neg(s(x1))), pos(s(x0)), neg(s(x1)))
MINUS(neg(s(x0)), pos(s(x1))) → COND(if(true, neg(s(x0)), pos(s(x1))), neg(s(x0)), pos(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(if(false, neg(0), neg(s(x0))), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(if(false, neg(0), neg(0)), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(if(false, neg(0), pos(0)), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(neg(0), pos(s(x0))) → COND(if(true, neg(0), pos(s(x0))), neg(0), pos(s(x0)))
MINUS(neg(s(x0)), pos(0)) → COND(if(true, neg(s(x0)), pos(0)), neg(s(x0)), pos(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(if(false, pos(s(x0)), neg(s(x1))), pos(s(x0)), neg(s(x1)))
MINUS(neg(s(x0)), pos(s(x1))) → COND(if(true, neg(s(x0)), pos(s(x1))), neg(s(x0)), pos(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(if(false, neg(0), neg(s(x0))), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(if(false, neg(0), neg(0)), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(if(false, neg(0), pos(0)), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(neg(0), pos(s(x0))) → COND(if(true, neg(0), pos(s(x0))), neg(0), pos(s(x0)))
MINUS(neg(s(x0)), pos(0)) → COND(if(true, neg(s(x0)), pos(0)), neg(s(x0)), pos(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(if(false, pos(s(x0)), neg(s(x1))), pos(s(x0)), neg(s(x1)))
MINUS(neg(s(x0)), pos(s(x1))) → COND(if(true, neg(s(x0)), pos(s(x1))), neg(s(x0)), pos(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(if(false, neg(0), neg(s(x0))), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(if(false, neg(0), neg(0)), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(if(false, neg(0), pos(0)), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(neg(0), pos(s(x0))) → COND(if(true, neg(0), pos(s(x0))), neg(0), pos(s(x0))) at position [0] we obtained the following new rules [LPAR04]:

MINUS(neg(0), pos(s(x0))) → COND(neg(0), neg(0), pos(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(neg(s(x0)), pos(0)) → COND(if(true, neg(s(x0)), pos(0)), neg(s(x0)), pos(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(if(false, pos(s(x0)), neg(s(x1))), pos(s(x0)), neg(s(x1)))
MINUS(neg(s(x0)), pos(s(x1))) → COND(if(true, neg(s(x0)), pos(s(x1))), neg(s(x0)), pos(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(if(false, neg(0), neg(s(x0))), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(if(false, neg(0), neg(0)), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(if(false, neg(0), pos(0)), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(neg(0), pos(s(x0))) → COND(neg(0), neg(0), pos(s(x0)))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                ↳ UsableRulesProof

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

MINUS(neg(s(x0)), pos(0)) → COND(if(true, neg(s(x0)), pos(0)), neg(s(x0)), pos(0))
COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(pos(s(x0)), neg(s(x1))) → COND(if(false, pos(s(x0)), neg(s(x1))), pos(s(x0)), neg(s(x1)))
MINUS(neg(s(x0)), pos(s(x1))) → COND(if(true, neg(s(x0)), pos(s(x1))), neg(s(x0)), pos(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(if(false, neg(0), neg(s(x0))), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(if(false, neg(0), neg(0)), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(if(false, neg(0), pos(0)), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(neg(s(x0)), pos(0)) → COND(if(true, neg(s(x0)), pos(0)), neg(s(x0)), pos(0)) at position [0] we obtained the following new rules [LPAR04]:

MINUS(neg(s(x0)), pos(0)) → COND(neg(s(x0)), neg(s(x0)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(pos(s(x0)), neg(s(x1))) → COND(if(false, pos(s(x0)), neg(s(x1))), pos(s(x0)), neg(s(x1)))
MINUS(neg(s(x0)), pos(s(x1))) → COND(if(true, neg(s(x0)), pos(s(x1))), neg(s(x0)), pos(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(if(false, neg(0), neg(s(x0))), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(if(false, neg(0), neg(0)), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(if(false, neg(0), pos(0)), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(neg(s(x0)), pos(0)) → COND(neg(s(x0)), neg(s(x0)), pos(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                ↳ UsableRulesProof

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

MINUS(pos(s(x0)), neg(s(x1))) → COND(if(false, pos(s(x0)), neg(s(x1))), pos(s(x0)), neg(s(x1)))
COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(neg(s(x0)), pos(s(x1))) → COND(if(true, neg(s(x0)), pos(s(x1))), neg(s(x0)), pos(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(if(false, neg(0), neg(s(x0))), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(if(false, neg(0), neg(0)), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(if(false, neg(0), pos(0)), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(pos(s(x0)), neg(s(x1))) → COND(if(false, pos(s(x0)), neg(s(x1))), pos(s(x0)), neg(s(x1))) at position [0] we obtained the following new rules [LPAR04]:

MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(neg(s(x0)), pos(s(x1))) → COND(if(true, neg(s(x0)), pos(s(x1))), neg(s(x0)), pos(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(if(false, neg(0), neg(s(x0))), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(if(false, neg(0), neg(0)), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(if(false, neg(0), pos(0)), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(neg(s(x0)), pos(s(x1))) → COND(if(true, neg(s(x0)), pos(s(x1))), neg(s(x0)), pos(s(x1))) at position [0] we obtained the following new rules [LPAR04]:

MINUS(neg(s(x0)), pos(s(x1))) → COND(neg(s(x0)), neg(s(x0)), pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(neg(0), neg(s(x0))) → COND(if(false, neg(0), neg(s(x0))), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(if(false, neg(0), neg(0)), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(if(false, neg(0), pos(0)), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(s(x0)), pos(s(x1))) → COND(neg(s(x0)), neg(s(x0)), pos(s(x1)))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                ↳ UsableRulesProof

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

MINUS(neg(0), neg(s(x0))) → COND(if(false, neg(0), neg(s(x0))), neg(0), neg(s(x0)))
COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(neg(0), neg(0)) → COND(if(false, neg(0), neg(0)), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(if(false, neg(0), pos(0)), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(neg(0), neg(s(x0))) → COND(if(false, neg(0), neg(s(x0))), neg(0), neg(s(x0))) at position [0] we obtained the following new rules [LPAR04]:

MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(neg(0), neg(0)) → COND(if(false, neg(0), neg(0)), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(if(false, neg(0), pos(0)), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(neg(0), neg(0)) → COND(if(false, neg(0), neg(0)), neg(0), neg(0)) at position [0] we obtained the following new rules [LPAR04]:

MINUS(neg(0), neg(0)) → COND(neg(0), neg(0), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(neg(0), pos(0)) → COND(if(false, neg(0), pos(0)), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(neg(0), neg(0), neg(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(neg(0), pos(0)) → COND(if(false, neg(0), pos(0)), neg(0), pos(0)) at position [0] we obtained the following new rules [LPAR04]:

MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(neg(0), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(pos(0), neg(0)) → COND(if(false, pos(0), neg(0)), pos(0), neg(0)) at position [0] we obtained the following new rules [LPAR04]:

MINUS(pos(0), neg(0)) → COND(neg(0), pos(0), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(neg(0), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(neg(0), pos(0), neg(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(pos(s(x0)), pos(0)) → COND(if(false, pos(s(x0)), pos(0)), pos(s(x0)), pos(0)) at position [0] we obtained the following new rules [LPAR04]:

MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(neg(0), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(neg(0), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(pos(0), pos(0)) → COND(if(false, pos(0), pos(0)), pos(0), pos(0)) at position [0] we obtained the following new rules [LPAR04]:

MINUS(pos(0), pos(0)) → COND(pos(0), pos(0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(neg(0), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(neg(0), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(pos(0), pos(0), pos(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(pos(0), pos(s(x0))) → COND(if(true, pos(0), pos(s(x0))), pos(0), pos(s(x0))) at position [0] we obtained the following new rules [LPAR04]:

MINUS(pos(0), pos(s(x0))) → COND(pos(0), pos(0), pos(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(neg(0), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(neg(0), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(pos(0), pos(0), pos(0))
MINUS(pos(0), pos(s(x0))) → COND(pos(0), pos(0), pos(s(x0)))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                ↳ UsableRulesProof

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

MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(neg(0), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(neg(0), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(pos(0), pos(0), pos(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(neg(s(x0)), neg(0)) → COND(if(true, neg(s(x0)), neg(0)), neg(s(x0)), neg(0)) at position [0] we obtained the following new rules [LPAR04]:

MINUS(neg(s(x0)), neg(0)) → COND(neg(s(x0)), neg(s(x0)), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(neg(0), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(neg(0), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(pos(0), pos(0), pos(0))
MINUS(neg(s(x0)), neg(0)) → COND(neg(s(x0)), neg(s(x0)), neg(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(neg(0), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(neg(0), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(pos(0), pos(0), pos(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(pos(0), neg(s(x0))) → COND(if(false, pos(0), neg(s(x0))), pos(0), neg(s(x0))) at position [0] we obtained the following new rules [LPAR04]:

MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(neg(0), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(neg(0), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(pos(0), pos(0), pos(0))
MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 MINUS(pos(s(x0)), neg(0)) → COND(if(false, pos(s(x0)), neg(0)), pos(s(x0)), neg(0)) at position [0] we obtained the following new rules [LPAR04]:

MINUS(pos(s(x0)), neg(0)) → COND(neg(0), pos(s(x0)), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                ↳ UsableRulesProof

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

COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(neg(0), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(neg(0), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(pos(0), pos(0), pos(0))
MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(neg(0), pos(s(x0)), neg(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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(y, x, y) → MINUS(x, plus_int(pos(s(0)), y)) at position [1] we obtained the following new rules [LPAR04]:

COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(plus_nat(s(0), x1)))
COND(neg(x1), y1, neg(x1)) → MINUS(y1, minus_nat(s(0), x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(neg(0), neg(0)) → COND(neg(0), neg(0), neg(0))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(0), neg(0)) → COND(neg(0), pos(0), neg(0))
MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(pos(0), pos(0), pos(0))
MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))
MINUS(pos(s(x0)), neg(0)) → COND(neg(0), pos(s(x0)), neg(0))
COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(plus_nat(s(0), x1)))
COND(neg(x1), y1, neg(x1)) → MINUS(y1, minus_nat(s(0), x1))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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 with 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(plus_nat(s(0), x1)))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(pos(0), pos(0), pos(0))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(plus_nat(s(0), x1)))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(pos(0), pos(0), pos(0))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Rewriting

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(plus_nat(s(0), x1)))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(pos(0), pos(0), pos(0))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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 rewriting [LPAR04] the rule COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(plus_nat(s(0), x1))) at position [1,0] we obtained the following new rules [LPAR04]:

COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(s(plus_nat(0, x1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Rewriting

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
MINUS(neg(0), pos(0)) → COND(pos(0), neg(0), pos(0))
MINUS(pos(s(x0)), pos(0)) → COND(pos(0), pos(s(x0)), pos(0))
MINUS(pos(0), pos(0)) → COND(pos(0), pos(0), pos(0))
COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(s(plus_nat(0, x1))))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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 1 SCC with 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Rewriting

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(s(plus_nat(0, x1))))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))

The TRS R consists of the following rules:

plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Rewriting

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Rewriting

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(s(plus_nat(0, x1))))
MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))

The TRS R consists of the following rules:

plus_nat(0, x) → x
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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 rewriting [LPAR04] the rule COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(s(plus_nat(0, x1)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Rewriting

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Rewriting

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(s(x1)))

The TRS R consists of the following rules:

plus_nat(0, x) → x
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Rewriting

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Rewriting

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QReductionProof

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(s(x1)))

The TRS R consists of the following rules:

less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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 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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Rewriting

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Rewriting

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ Instantiation

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
COND(pos(x1), y1, pos(x1)) → MINUS(y1, pos(s(x1)))

The TRS R consists of the following rules:

less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

COND(pos(s(z1)), pos(s(z0)), pos(s(z1))) → MINUS(pos(s(z0)), pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Rewriting

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Rewriting

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1)))
COND(pos(s(z1)), pos(s(z0)), pos(s(z1))) → MINUS(pos(s(z0)), pos(s(s(z1))))

The TRS R consists of the following rules:

less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule MINUS(pos(s(x0)), pos(s(x1))) → COND(if(less_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))), pos(s(x0)), pos(s(x1))) we obtained the following new rules [LPAR04]:

MINUS(pos(s(z1)), pos(s(s(z0)))) → COND(if(less_int(pos(z1), pos(s(z0))), pos(s(z1)), pos(s(s(z0)))), pos(s(z1)), pos(s(s(z0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Rewriting

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Rewriting

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Instantiation

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND(pos(s(z1)), pos(s(z0)), pos(s(z1))) → MINUS(pos(s(z0)), pos(s(s(z1))))
MINUS(pos(s(z1)), pos(s(s(z0)))) → COND(if(less_int(pos(z1), pos(s(z0))), pos(s(z1)), pos(s(s(z0)))), pos(s(z1)), pos(s(s(z0))))

The TRS R consists of the following rules:

less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

COND(pos(s(s(z1))), pos(s(z0)), pos(s(s(z1)))) → MINUS(pos(s(z0)), pos(s(s(s(z1)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Rewriting

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Rewriting

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Instantiation

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(s(z1)), pos(s(s(z0)))) → COND(if(less_int(pos(z1), pos(s(z0))), pos(s(z1)), pos(s(s(z0)))), pos(s(z1)), pos(s(s(z0))))
COND(pos(s(s(z1))), pos(s(z0)), pos(s(s(z1)))) → MINUS(pos(s(z0)), pos(s(s(s(z1)))))

The TRS R consists of the following rules:

less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

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

MINUS(pos(s(z1)), pos(s(s(s(z0))))) → COND(if(less_int(pos(z1), pos(s(s(z0)))), pos(s(z1)), pos(s(s(s(z0))))), pos(s(z1)), pos(s(s(s(z0)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Rewriting

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Rewriting

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND(pos(s(s(z1))), pos(s(z0)), pos(s(s(z1)))) → MINUS(pos(s(z0)), pos(s(s(s(z1)))))
MINUS(pos(s(z1)), pos(s(s(s(z0))))) → COND(if(less_int(pos(z1), pos(s(s(z0)))), pos(s(z1)), pos(s(s(s(z0))))), pos(s(z1)), pos(s(s(s(z0)))))

The TRS R consists of the following rules:

less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Rewriting

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Rewriting

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ MNOCProof

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND(pos(s(s(z1))), pos(s(z0)), pos(s(s(z1)))) → MINUS(pos(s(z0)), pos(s(s(s(z1)))))
MINUS(pos(s(z1)), pos(s(s(s(z0))))) → COND(if(less_int(pos(z1), pos(s(s(z0)))), pos(s(z1)), pos(s(s(s(z0))))), pos(s(z1)), pos(s(s(s(z0)))))

The TRS R consists of the following rules:

less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), pos(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(pos(s(x0)), pos(s(x1)))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Rewriting

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Rewriting

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Instantiation

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Instantiation

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ MNOCProof

                                                                                                                                                                                        ↳ QDP

                                                                                                                                ↳ QDP

                ↳ UsableRulesProof

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

COND(pos(s(s(z1))), pos(s(z0)), pos(s(s(z1)))) → MINUS(pos(s(z0)), pos(s(s(s(z1)))))
MINUS(pos(s(z1)), pos(s(s(s(z0))))) → COND(if(less_int(pos(z1), pos(s(s(z0)))), pos(s(z1)), pos(s(s(s(z0))))), pos(s(z1)), pos(s(s(s(z0)))))

The TRS R consists of the following rules:

less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
if(false, u, v) → v
if(true, u, v) → u

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                ↳ UsableRulesProof

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

MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
COND(neg(x1), y1, neg(x1)) → MINUS(y1, minus_nat(s(0), x1))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))

The TRS R consists of the following rules:

if(false, u, v) → v
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(true, u, v) → u
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)
less_int(pos(0), pos(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(pos(s(x)), pos(0)) → false
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                ↳ UsableRulesProof

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

MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
COND(neg(x1), y1, neg(x1)) → MINUS(y1, minus_nat(s(0), x1))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))

The TRS R consists of the following rules:

less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                ↳ UsableRulesProof

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

MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
COND(neg(x1), y1, neg(x1)) → MINUS(y1, minus_nat(s(0), x1))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))

The TRS R consists of the following rules:

less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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 instantiating [LPAR04] the rule COND(neg(x1), y1, neg(x1)) → MINUS(y1, minus_nat(s(0), x1)) we obtained the following new rules [LPAR04]:

COND(neg(s(z0)), pos(0), neg(s(z0))) → MINUS(pos(0), minus_nat(s(0), s(z0)))
COND(neg(s(z1)), pos(s(z0)), neg(s(z1))) → MINUS(pos(s(z0)), minus_nat(s(0), s(z1)))
COND(neg(s(z1)), neg(s(z0)), neg(s(z1))) → MINUS(neg(s(z0)), minus_nat(s(0), s(z1)))
COND(neg(s(z0)), neg(0), neg(s(z0))) → MINUS(neg(0), minus_nat(s(0), s(z0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))
COND(neg(s(z0)), pos(0), neg(s(z0))) → MINUS(pos(0), minus_nat(s(0), s(z0)))
COND(neg(s(z1)), pos(s(z0)), neg(s(z1))) → MINUS(pos(s(z0)), minus_nat(s(0), s(z1)))
COND(neg(s(z1)), neg(s(z0)), neg(s(z1))) → MINUS(neg(s(z0)), minus_nat(s(0), s(z1)))
COND(neg(s(z0)), neg(0), neg(s(z0))) → MINUS(neg(0), minus_nat(s(0), s(z0)))

The TRS R consists of the following rules:

less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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 4 SCCs.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(z0)), pos(0), neg(s(z0))) → MINUS(pos(0), minus_nat(s(0), s(z0)))
MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))

The TRS R consists of the following rules:

less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(z0)), pos(0), neg(s(z0))) → MINUS(pos(0), minus_nat(s(0), s(z0)))
MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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].

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(z0)), pos(0), neg(s(z0))) → MINUS(pos(0), minus_nat(s(0), s(z0)))
MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))

The TRS R consists of the following rules:

minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
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 rewriting [LPAR04] the rule COND(neg(s(z0)), pos(0), neg(s(z0))) → MINUS(pos(0), minus_nat(s(0), s(z0))) at position [1] we obtained the following new rules [LPAR04]:

COND(neg(s(z0)), pos(0), neg(s(z0))) → MINUS(pos(0), minus_nat(0, z0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))
COND(neg(s(z0)), pos(0), neg(s(z0))) → MINUS(pos(0), minus_nat(0, z0))

The TRS R consists of the following rules:

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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))
COND(neg(s(z0)), pos(0), neg(s(z0))) → MINUS(pos(0), minus_nat(0, z0))

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(neg(s(z0)), pos(0), neg(s(z0))) → MINUS(pos(0), minus_nat(0, z0)) at position [1] we obtained the following new rules [LPAR04]:

COND(neg(s(s(x0))), pos(0), neg(s(s(x0)))) → MINUS(pos(0), neg(s(x0)))
COND(neg(s(0)), pos(0), neg(s(0))) → MINUS(pos(0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))
COND(neg(s(s(x0))), pos(0), neg(s(s(x0)))) → MINUS(pos(0), neg(s(x0)))
COND(neg(s(0)), pos(0), neg(s(0))) → MINUS(pos(0), 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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), pos(0), neg(s(s(x0)))) → MINUS(pos(0), neg(s(x0)))
MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))

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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), pos(0), neg(s(s(x0)))) → MINUS(pos(0), neg(s(x0)))
MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))

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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), pos(0), neg(s(s(x0)))) → MINUS(pos(0), neg(s(x0)))
MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0)))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule MINUS(pos(0), neg(s(x0))) → COND(neg(s(x0)), pos(0), neg(s(x0))) we obtained the following new rules [LPAR04]:

MINUS(pos(0), neg(s(s(y_0)))) → COND(neg(s(s(y_0))), pos(0), neg(s(s(y_0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), pos(0), neg(s(s(x0)))) → MINUS(pos(0), neg(s(x0)))
MINUS(pos(0), neg(s(s(y_0)))) → COND(neg(s(s(y_0))), pos(0), neg(s(s(y_0))))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND(neg(s(s(x0))), pos(0), neg(s(s(x0)))) → MINUS(pos(0), neg(s(x0))) we obtained the following new rules [LPAR04]:

COND(neg(s(s(s(y_0)))), pos(0), neg(s(s(s(y_0))))) → MINUS(pos(0), neg(s(s(y_0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ MRRProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(0), neg(s(s(y_0)))) → COND(neg(s(s(y_0))), pos(0), neg(s(s(y_0))))
COND(neg(s(s(s(y_0)))), pos(0), neg(s(s(s(y_0))))) → MINUS(pos(0), neg(s(s(y_0))))

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:

COND(neg(s(s(s(y_0)))), pos(0), neg(s(s(s(y_0))))) → MINUS(pos(0), neg(s(s(y_0))))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1
POL(COND(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + x₃
POL(MINUS(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(neg(x₁)) = 2·x₁
POL(pos(x₁)) = x₁
POL(s(x₁)) = 2 + x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ MRRProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(0), neg(s(s(y_0)))) → COND(neg(s(s(y_0))), pos(0), neg(s(s(y_0))))

R is empty.
Q is empty.
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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(z0)), neg(0), neg(s(z0))) → MINUS(neg(0), minus_nat(s(0), s(z0)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))

The TRS R consists of the following rules:

less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(z0)), neg(0), neg(s(z0))) → MINUS(neg(0), minus_nat(s(0), s(z0)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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].

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(z0)), neg(0), neg(s(z0))) → MINUS(neg(0), minus_nat(s(0), s(z0)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))

The TRS R consists of the following rules:

minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
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 rewriting [LPAR04] the rule COND(neg(s(z0)), neg(0), neg(s(z0))) → MINUS(neg(0), minus_nat(s(0), s(z0))) at position [1] we obtained the following new rules [LPAR04]:

COND(neg(s(z0)), neg(0), neg(s(z0))) → MINUS(neg(0), minus_nat(0, z0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
COND(neg(s(z0)), neg(0), neg(s(z0))) → MINUS(neg(0), minus_nat(0, z0))

The TRS R consists of the following rules:

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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
COND(neg(s(z0)), neg(0), neg(s(z0))) → MINUS(neg(0), minus_nat(0, z0))

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(neg(s(z0)), neg(0), neg(s(z0))) → MINUS(neg(0), minus_nat(0, z0)) at position [1] we obtained the following new rules [LPAR04]:

COND(neg(s(s(x0))), neg(0), neg(s(s(x0)))) → MINUS(neg(0), neg(s(x0)))
COND(neg(s(0)), neg(0), neg(s(0))) → MINUS(neg(0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))
COND(neg(s(s(x0))), neg(0), neg(s(s(x0)))) → MINUS(neg(0), neg(s(x0)))
COND(neg(s(0)), neg(0), neg(s(0))) → MINUS(neg(0), 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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), neg(0), neg(s(s(x0)))) → MINUS(neg(0), neg(s(x0)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))

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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), neg(0), neg(s(s(x0)))) → MINUS(neg(0), neg(s(x0)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))

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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), neg(0), neg(s(s(x0)))) → MINUS(neg(0), neg(s(x0)))
MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0)))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule MINUS(neg(0), neg(s(x0))) → COND(neg(s(x0)), neg(0), neg(s(x0))) we obtained the following new rules [LPAR04]:

MINUS(neg(0), neg(s(s(y_0)))) → COND(neg(s(s(y_0))), neg(0), neg(s(s(y_0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), neg(0), neg(s(s(x0)))) → MINUS(neg(0), neg(s(x0)))
MINUS(neg(0), neg(s(s(y_0)))) → COND(neg(s(s(y_0))), neg(0), neg(s(s(y_0))))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND(neg(s(s(x0))), neg(0), neg(s(s(x0)))) → MINUS(neg(0), neg(s(x0))) we obtained the following new rules [LPAR04]:

COND(neg(s(s(s(y_0)))), neg(0), neg(s(s(s(y_0))))) → MINUS(neg(0), neg(s(s(y_0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ MRRProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

MINUS(neg(0), neg(s(s(y_0)))) → COND(neg(s(s(y_0))), neg(0), neg(s(s(y_0))))
COND(neg(s(s(s(y_0)))), neg(0), neg(s(s(s(y_0))))) → MINUS(neg(0), neg(s(s(y_0))))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 2
POL(COND(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(MINUS(x₁, x₂)) = 2·x₁ + 2·x₂
POL(neg(x₁)) = 2 + 2·x₁
POL(s(x₁)) = 2 + 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ MRRProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ PisEmptyProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(z1)), pos(s(z0)), neg(s(z1))) → MINUS(pos(s(z0)), minus_nat(s(0), s(z1)))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(z1)), pos(s(z0)), neg(s(z1))) → MINUS(pos(s(z0)), minus_nat(s(0), s(z1)))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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].

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(z1)), pos(s(z0)), neg(s(z1))) → MINUS(pos(s(z0)), minus_nat(s(0), s(z1)))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
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 rewriting [LPAR04] the rule COND(neg(s(z1)), pos(s(z0)), neg(s(z1))) → MINUS(pos(s(z0)), minus_nat(s(0), s(z1))) at position [1] we obtained the following new rules [LPAR04]:

COND(neg(s(z1)), pos(s(z0)), neg(s(z1))) → MINUS(pos(s(z0)), minus_nat(0, z1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
COND(neg(s(z1)), pos(s(z0)), neg(s(z1))) → MINUS(pos(s(z0)), minus_nat(0, z1))

The TRS R consists of the following rules:

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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
COND(neg(s(z1)), pos(s(z0)), neg(s(z1))) → MINUS(pos(s(z0)), minus_nat(0, z1))

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(neg(s(z1)), pos(s(z0)), neg(s(z1))) → MINUS(pos(s(z0)), minus_nat(0, z1)) at position [1] we obtained the following new rules [LPAR04]:

COND(neg(s(0)), pos(s(y1)), neg(s(0))) → MINUS(pos(s(y1)), pos(0))
COND(neg(s(s(x0))), pos(s(y1)), neg(s(s(x0)))) → MINUS(pos(s(y1)), neg(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ 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:

MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1)))
COND(neg(s(0)), pos(s(y1)), neg(s(0))) → MINUS(pos(s(y1)), pos(0))
COND(neg(s(s(x0))), pos(s(y1)), neg(s(s(x0)))) → MINUS(pos(s(y1)), neg(s(x0)))

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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), pos(s(y1)), neg(s(s(x0)))) → MINUS(pos(s(y1)), neg(s(x0)))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), 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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), pos(s(y1)), neg(s(s(x0)))) → MINUS(pos(s(y1)), neg(s(x0)))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), 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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), pos(s(y1)), neg(s(s(x0)))) → MINUS(pos(s(y1)), neg(s(x0)))
MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), 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 MINUS(pos(s(x0)), neg(s(x1))) → COND(neg(s(x1)), pos(s(x0)), neg(s(x1))) we obtained the following new rules [LPAR04]:

MINUS(pos(s(x0)), neg(s(s(y_0)))) → COND(neg(s(s(y_0))), pos(s(x0)), neg(s(s(y_0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), pos(s(y1)), neg(s(s(x0)))) → MINUS(pos(s(y1)), neg(s(x0)))
MINUS(pos(s(x0)), neg(s(s(y_0)))) → COND(neg(s(s(y_0))), pos(s(x0)), neg(s(s(y_0))))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND(neg(s(s(x0))), pos(s(y1)), neg(s(s(x0)))) → MINUS(pos(s(y1)), neg(s(x0))) we obtained the following new rules [LPAR04]:

COND(neg(s(s(s(y_1)))), pos(s(x1)), neg(s(s(s(y_1))))) → MINUS(pos(s(x1)), neg(s(s(y_1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ MRRProof

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(s(x0)), neg(s(s(y_0)))) → COND(neg(s(s(y_0))), pos(s(x0)), neg(s(s(y_0))))
COND(neg(s(s(s(y_1)))), pos(s(x1)), neg(s(s(s(y_1))))) → MINUS(pos(s(x1)), neg(s(s(y_1))))

COND(neg(s(s(s(y_1)))), pos(s(x1)), neg(s(s(s(y_1))))) → MINUS(pos(s(x1)), neg(s(s(y_1))))

Used ordering: Polynomial interpretation [POLO]:

POL(COND(x₁, x₂, x₃)) = x₁ + 2·x₂ + x₃
POL(MINUS(x₁, x₂)) = 2·x₁ + 2·x₂
POL(neg(x₁)) = 2·x₁
POL(pos(x₁)) = 2·x₁
POL(s(x₁)) = 2 + 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ MRRProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                ↳ UsableRulesProof

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

MINUS(pos(s(x0)), neg(s(s(y_0)))) → COND(neg(s(s(y_0))), pos(s(x0)), neg(s(s(y_0))))

R is empty.
Q is empty.
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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                ↳ UsableRulesProof

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

COND(neg(s(z1)), neg(s(z0)), neg(s(z1))) → MINUS(neg(s(z0)), minus_nat(s(0), s(z1)))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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 rewriting [LPAR04] the rule COND(neg(s(z1)), neg(s(z0)), neg(s(z1))) → MINUS(neg(s(z0)), minus_nat(s(0), s(z1))) at position [1] we obtained the following new rules [LPAR04]:

COND(neg(s(z1)), neg(s(z0)), neg(s(z1))) → MINUS(neg(s(z0)), minus_nat(0, z1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                ↳ UsableRulesProof

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

MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
COND(neg(s(z1)), neg(s(z0)), neg(s(z1))) → MINUS(neg(s(z0)), minus_nat(0, z1))

The TRS R consists of the following rules:

less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                ↳ UsableRulesProof

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

MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
COND(neg(s(z1)), neg(s(z0)), neg(s(z1))) → MINUS(neg(s(z0)), minus_nat(0, z1))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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(neg(s(z1)), neg(s(z0)), neg(s(z1))) → MINUS(neg(s(z0)), minus_nat(0, z1)) at position [1] we obtained the following new rules [LPAR04]:

COND(neg(s(s(x0))), neg(s(y1)), neg(s(s(x0)))) → MINUS(neg(s(y1)), neg(s(x0)))
COND(neg(s(0)), neg(s(y1)), neg(s(0))) → MINUS(neg(s(y1)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))
COND(neg(s(s(x0))), neg(s(y1)), neg(s(s(x0)))) → MINUS(neg(s(y1)), neg(s(x0)))
COND(neg(s(0)), neg(s(y1)), neg(s(0))) → MINUS(neg(s(y1)), pos(0))

The TRS R consists of the following rules:

minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ UsableRulesProof

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), neg(s(y1)), neg(s(s(x0)))) → MINUS(neg(s(y1)), neg(s(x0)))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(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))
less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QReductionProof

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), neg(s(y1)), neg(s(s(x0)))) → MINUS(neg(s(y1)), neg(s(x0)))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                ↳ UsableRulesProof

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

COND(neg(s(s(x0))), neg(s(y1)), neg(s(s(x0)))) → MINUS(neg(s(y1)), neg(s(x0)))
MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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(neg(s(s(x0))), neg(s(y1)), neg(s(s(x0)))) → MINUS(neg(s(y1)), neg(s(x0)))

The remaining pairs can at least be oriented weakly.

MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND(x₁, x₂, x₃)) = x₃
POL(MINUS(x₁, x₂)) = x₂
POL(false) = 0
POL(if(x₁, x₂, x₃)) = 0
POL(less_int(x₁, x₂)) = 0
POL(neg(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

                        ↳ Rewriting

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Rewriting

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ AND

                                                                                                                                ↳ QDP

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QReductionProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                ↳ UsableRulesProof

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

MINUS(neg(s(x0)), neg(s(x1))) → COND(if(less_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

less_int(neg(0), neg(0)) → false
less_int(neg(0), neg(s(y))) → false
less_int(neg(s(x)), neg(0)) → true
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u

The set Q consists of the following terms:

if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_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.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

MINUS(x, y) → COND(min(x, y), x, y)
COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))

The TRS R consists of the following rules:

min(u, v) → if(less_int(u, v), u, v)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u
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:

min(x0, x1)
cond(x0, x1, x0)
if(false, x0, x1)
if(true, x0, x1)
minus(x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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].

cond(x0, x1, x0)
minus(x0, x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

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

MINUS(x, y) → COND(min(x, y), x, y)
COND(y, x, y) → MINUS(x, plus_int(pos(s(0)), y))

The TRS R consists of the following rules:

min(u, v) → if(less_int(u, v), u, v)
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
if(false, u, v) → v
if(true, u, v) → u
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:

min(x0, x1)
if(false, x0, x1)
if(true, x0, x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(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))
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.