Termination proof of /tmp/tmpyywwDB/A14.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:

cond2(FALSE, x, y) → +@z(1@z, diff(+@z(x, 1@z), y))
cond1(TRUE, x, y) → 0@z
cond2(TRUE, x, y) → +@z(1@z, diff(x, +@z(y, 1@z)))
diff(x, y) → cond1(=@z(x, y), x, y)
cond1(FALSE, x, y) → cond2(>@z(x, y), x, y)

The set Q consists of the following terms:

cond2(FALSE, x0, x1)
cond1(TRUE, x0, x1)
cond2(TRUE, x0, x1)
diff(x0, x1)
cond1(FALSE, 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:

cond2(false, x, y) → plus_int(pos(s(0)), diff(plus_int(pos(s(0)), x), y))
cond1(true, x, y) → pos(0)
cond2(true, x, y) → plus_int(pos(s(0)), diff(x, plus_int(pos(s(0)), y)))
diff(x, y) → cond1(equal_int(x, y), x, y)
cond1(false, x, y) → cond2(greater_int(x, y), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

COND2(false, x, y) → PLUS_INT(pos(s(0)), diff(plus_int(pos(s(0)), x), y))
COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
COND2(false, x, y) → PLUS_INT(pos(s(0)), x)
COND2(true, x, y) → PLUS_INT(pos(s(0)), diff(x, plus_int(pos(s(0)), y)))
COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y))
COND2(true, x, y) → PLUS_INT(pos(s(0)), y)
DIFF(x, y) → COND1(equal_int(x, y), x, y)
DIFF(x, y) → EQUAL_INT(x, y)
COND1(false, x, y) → COND2(greater_int(x, y), x, y)
COND1(false, x, y) → GREATER_INT(x, y)
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
EQUAL_INT(pos(s(x)), pos(s(y))) → EQUAL_INT(pos(x), pos(y))
EQUAL_INT(neg(s(x)), neg(s(y))) → EQUAL_INT(neg(x), neg(y))
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

The TRS R consists of the following rules:

cond2(false, x, y) → plus_int(pos(s(0)), diff(plus_int(pos(s(0)), x), y))
cond1(true, x, y) → pos(0)
cond2(true, x, y) → plus_int(pos(s(0)), diff(x, plus_int(pos(s(0)), y)))
diff(x, y) → cond1(equal_int(x, y), x, y)
cond1(false, x, y) → cond2(greater_int(x, y), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

COND2(false, x, y) → PLUS_INT(pos(s(0)), diff(plus_int(pos(s(0)), x), y))
COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
COND2(false, x, y) → PLUS_INT(pos(s(0)), x)
COND2(true, x, y) → PLUS_INT(pos(s(0)), diff(x, plus_int(pos(s(0)), y)))
COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y))
COND2(true, x, y) → PLUS_INT(pos(s(0)), y)
DIFF(x, y) → COND1(equal_int(x, y), x, y)
DIFF(x, y) → EQUAL_INT(x, y)
COND1(false, x, y) → COND2(greater_int(x, y), x, y)
COND1(false, x, y) → GREATER_INT(x, y)
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
EQUAL_INT(pos(s(x)), pos(s(y))) → EQUAL_INT(pos(x), pos(y))
EQUAL_INT(neg(s(x)), neg(s(y))) → EQUAL_INT(neg(x), neg(y))
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

The TRS R consists of the following rules:

cond2(false, x, y) → plus_int(pos(s(0)), diff(plus_int(pos(s(0)), x), y))
cond1(true, x, y) → pos(0)
cond2(true, x, y) → plus_int(pos(s(0)), diff(x, plus_int(pos(s(0)), y)))
diff(x, y) → cond1(equal_int(x, y), x, y)
cond1(false, x, y) → cond2(greater_int(x, y), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 10 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

cond2(false, x, y) → plus_int(pos(s(0)), diff(plus_int(pos(s(0)), x), y))
cond1(true, x, y) → pos(0)
cond2(true, x, y) → plus_int(pos(s(0)), diff(x, plus_int(pos(s(0)), y)))
diff(x, y) → cond1(equal_int(x, y), x, y)
cond1(false, x, y) → cond2(greater_int(x, y), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

The following dependency pairs can be deleted:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

cond2(false, x, y) → plus_int(pos(s(0)), diff(plus_int(pos(s(0)), x), y))
cond1(true, x, y) → pos(0)
cond2(true, x, y) → plus_int(pos(s(0)), diff(x, plus_int(pos(s(0)), y)))
diff(x, y) → cond1(equal_int(x, y), x, y)
cond1(false, x, y) → cond2(greater_int(x, y), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

cond2(false, x, y) → plus_int(pos(s(0)), diff(plus_int(pos(s(0)), x), y))
cond1(true, x, y) → pos(0)
cond2(true, x, y) → plus_int(pos(s(0)), diff(x, plus_int(pos(s(0)), y)))
diff(x, y) → cond1(equal_int(x, y), x, y)
cond1(false, x, y) → cond2(greater_int(x, y), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(EQUAL_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

              ↳ 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

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

cond2(false, x, y) → plus_int(pos(s(0)), diff(plus_int(pos(s(0)), x), y))
cond1(true, x, y) → pos(0)
cond2(true, x, y) → plus_int(pos(s(0)), diff(x, plus_int(pos(s(0)), y)))
diff(x, y) → cond1(equal_int(x, y), x, y)
cond1(false, x, y) → cond2(greater_int(x, y), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(EQUAL_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

              ↳ 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

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

cond2(false, x, y) → plus_int(pos(s(0)), diff(plus_int(pos(s(0)), x), y))
cond1(true, x, y) → pos(0)
cond2(true, x, y) → plus_int(pos(s(0)), diff(x, plus_int(pos(s(0)), y)))
diff(x, y) → cond1(equal_int(x, y), x, y)
cond1(false, x, y) → cond2(greater_int(x, y), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ 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:

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

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

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

cond2(false, x, y) → plus_int(pos(s(0)), diff(plus_int(pos(s(0)), x), y))
cond1(true, x, y) → pos(0)
cond2(true, x, y) → plus_int(pos(s(0)), diff(x, plus_int(pos(s(0)), y)))
diff(x, y) → cond1(equal_int(x, y), x, y)
cond1(false, x, y) → cond2(greater_int(x, y), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

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

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

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

COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
DIFF(x, y) → COND1(equal_int(x, y), x, y)
COND1(false, x, y) → COND2(greater_int(x, y), x, y)
COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y))

The TRS R consists of the following rules:

cond2(false, x, y) → plus_int(pos(s(0)), diff(plus_int(pos(s(0)), x), y))
cond1(true, x, y) → pos(0)
cond2(true, x, y) → plus_int(pos(s(0)), diff(x, plus_int(pos(s(0)), y)))
diff(x, y) → cond1(equal_int(x, y), x, y)
cond1(false, x, y) → cond2(greater_int(x, y), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
DIFF(x, y) → COND1(equal_int(x, y), x, y)
COND1(false, x, y) → COND2(greater_int(x, y), x, y)
COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y))

The TRS R consists of the following rules:

equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_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:

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

cond2(false, x0, x1)
cond1(true, x0, x1)
cond2(true, x0, x1)
diff(x0, x1)
cond1(false, x0, x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

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

COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
DIFF(x, y) → COND1(equal_int(x, y), x, y)
COND1(false, x, y) → COND2(greater_int(x, y), x, y)
COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y))

The TRS R consists of the following rules:

equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_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:

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

We have to consider all minimal (P,Q,R)-chains.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
Positions in right side of the pair:

[0,0]

Pair: COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y))
Positions in right side of the pair:

[1,0]

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                        ↳ RemovalProof

                        ↳ Narrowing

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

COND2(false, x, y, x_removed) → DIFF(plus_int(x_removed, x), y, x_removed)
DIFF(x, y, x_removed) → COND1(equal_int(x, y), x, y, x_removed)
COND1(false, x, y, x_removed) → COND2(greater_int(x, y), x, y, x_removed)
COND2(true, x, y, x_removed) → DIFF(x, plus_int(x_removed, y), x_removed)

The TRS R consists of the following rules:

equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_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:

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

[0,0]

Pair: COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y))
Positions in right side of the pair:

[1,0]

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                          ↳ QDP

                        ↳ Narrowing

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

COND2(false, x, y, x_removed) → DIFF(plus_int(x_removed, x), y, x_removed)
DIFF(x, y, x_removed) → COND1(equal_int(x, y), x, y, x_removed)
COND1(false, x, y, x_removed) → COND2(greater_int(x, y), x, y, x_removed)
COND2(true, x, y, x_removed) → DIFF(x, plus_int(x_removed, y), x_removed)

The TRS R consists of the following rules:

equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_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:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule DIFF(x, y) → COND1(equal_int(x, y), x, y) at position [0] we obtained the following new rules [LPAR04]:

DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(pos(0), neg(0)) → COND1(true, pos(0), neg(0))
DIFF(neg(0), pos(0)) → COND1(true, neg(0), pos(0))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
DIFF(neg(0), neg(0)) → COND1(true, neg(0), neg(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
DIFF(pos(0), pos(0)) → COND1(true, pos(0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

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

COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
COND1(false, x, y) → COND2(greater_int(x, y), x, y)
COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(pos(0), neg(0)) → COND1(true, pos(0), neg(0))
DIFF(neg(0), pos(0)) → COND1(true, neg(0), pos(0))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
DIFF(neg(0), neg(0)) → COND1(true, neg(0), neg(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
DIFF(pos(0), pos(0)) → COND1(true, pos(0), pos(0))

The TRS R consists of the following rules:

equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_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:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

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

DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
COND1(false, x, y) → COND2(greater_int(x, y), x, y)
COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y))

The TRS R consists of the following rules:

equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_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:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

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

DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
COND1(false, x, y) → COND2(greater_int(x, y), x, y)
COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND1(false, x, y) → COND2(greater_int(x, y), x, y) at position [0] we obtained the following new rules [LPAR04]:

COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
COND1(false, neg(0), pos(0)) → COND2(false, neg(0), pos(0))
COND1(false, pos(0), neg(0)) → COND2(false, pos(0), neg(0))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
COND1(false, neg(0), neg(0)) → COND2(false, neg(0), neg(0))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
COND1(false, pos(0), pos(0)) → COND2(false, pos(0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

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

DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
COND1(false, neg(0), pos(0)) → COND2(false, neg(0), pos(0))
COND1(false, pos(0), neg(0)) → COND2(false, pos(0), neg(0))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
COND1(false, neg(0), neg(0)) → COND2(false, neg(0), neg(0))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
COND1(false, pos(0), pos(0)) → COND2(false, pos(0), pos(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)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
equal_int(neg(0), neg(0)) → true
equal_int(neg(0), neg(s(y))) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND2(true, x, y) → DIFF(x, plus_int(pos(s(0)), y)) at position [1] we obtained the following new rules [LPAR04]:

COND2(true, y0, neg(x1)) → DIFF(y0, minus_nat(s(0), x1))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(plus_nat(s(0), x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, neg(x1)) → DIFF(y0, minus_nat(s(0), x1))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(plus_nat(s(0), x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND2(true, y0, pos(x1)) → DIFF(y0, pos(plus_nat(s(0), x1))) at position [1,0] we obtained the following new rules [LPAR04]:

COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(plus_nat(0, x1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, neg(x1)) → DIFF(y0, minus_nat(s(0), x1))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(plus_nat(0, x1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(plus_nat(0, x1)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y)
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, neg(x1)) → DIFF(y0, minus_nat(s(0), x1))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND2(false, x, y) → DIFF(plus_int(pos(s(0)), x), y) at position [0] we obtained the following new rules [LPAR04]:

COND2(false, pos(x1), y1) → DIFF(pos(plus_nat(s(0), x1)), y1)
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, neg(x1)) → DIFF(y0, minus_nat(s(0), x1))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
COND2(false, pos(x1), y1) → DIFF(pos(plus_nat(s(0), x1)), y1)
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, neg(x1)) → DIFF(y0, minus_nat(s(0), x1))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
COND2(false, pos(x1), y1) → DIFF(pos(plus_nat(s(0), x1)), y1)
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, neg(x1)) → DIFF(y0, minus_nat(s(0), x1))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
COND2(false, pos(x1), y1) → DIFF(pos(plus_nat(s(0), x1)), y1)
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND2(false, pos(x1), y1) → DIFF(pos(plus_nat(s(0), x1)), y1) at position [0,0] we obtained the following new rules [LPAR04]:

COND2(false, pos(x1), y1) → DIFF(pos(s(plus_nat(0, x1))), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, neg(x1)) → DIFF(y0, minus_nat(s(0), x1))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
COND2(false, pos(x1), y1) → DIFF(pos(s(plus_nat(0, x1))), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, neg(x1)) → DIFF(y0, minus_nat(s(0), x1))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
COND2(false, pos(x1), y1) → DIFF(pos(s(plus_nat(0, x1))), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND2(false, pos(x1), y1) → DIFF(pos(s(plus_nat(0, x1))), y1) at position [0,0,0] we obtained the following new rules [LPAR04]:

COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, neg(x1)) → DIFF(y0, minus_nat(s(0), x1))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, neg(x1)) → DIFF(y0, minus_nat(s(0), x1))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, neg(x1)) → DIFF(y0, minus_nat(s(0), x1))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)

The TRS R consists of the following rules:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND2(true, y0, neg(x1)) → DIFF(y0, minus_nat(s(0), x1)) we obtained the following new rules [LPAR04]:

COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), minus_nat(s(0), 0))
COND2(true, pos(0), neg(s(z0))) → DIFF(pos(0), minus_nat(s(0), s(z0)))
COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(s(0), s(z1)))
COND2(true, neg(0), neg(s(z0))) → DIFF(neg(0), minus_nat(s(0), s(z0)))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(s(0), s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), minus_nat(s(0), 0))
COND2(true, pos(0), neg(s(z0))) → DIFF(pos(0), minus_nat(s(0), s(z0)))
COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(s(0), s(z1)))
COND2(true, neg(0), neg(s(z0))) → DIFF(neg(0), minus_nat(s(0), s(z0)))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(s(0), s(z1)))

The TRS R consists of the following rules:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                    ↳ QDP

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

COND2(true, pos(0), neg(s(z0))) → DIFF(pos(0), minus_nat(s(0), s(z0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), minus_nat(s(0), 0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(s(0), s(z1)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(s(0), s(z1)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))

The TRS R consists of the following rules:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND2(true, pos(0), neg(s(z0))) → DIFF(pos(0), minus_nat(s(0), s(z0))) at position [1] we obtained the following new rules [LPAR04]:

COND2(true, pos(0), neg(s(z0))) → DIFF(pos(0), minus_nat(0, z0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                    ↳ QDP

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

DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), minus_nat(s(0), 0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(s(0), s(z1)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(s(0), s(z1)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
COND2(true, pos(0), neg(s(z0))) → DIFF(pos(0), minus_nat(0, z0))

The TRS R consists of the following rules:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
COND2(true, pos(0), neg(s(z0))) → DIFF(pos(0), minus_nat(0, z0))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))

The TRS R consists of the following rules:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QReductionProof

                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
COND2(true, pos(0), neg(s(z0))) → DIFF(pos(0), minus_nat(0, z0))
DIFF(pos(0), neg(s(x0))) → COND1(false, 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))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QReductionProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
COND2(true, pos(0), neg(s(z0))) → DIFF(pos(0), minus_nat(0, z0))
DIFF(pos(0), neg(s(x0))) → COND1(false, 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))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND2(true, pos(0), neg(s(z0))) → DIFF(pos(0), minus_nat(0, z0)) at position [1] we obtained the following new rules [LPAR04]:

COND2(true, pos(0), neg(s(0))) → DIFF(pos(0), pos(0))
COND2(true, pos(0), neg(s(s(x0)))) → DIFF(pos(0), neg(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QReductionProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
COND2(true, pos(0), neg(s(0))) → DIFF(pos(0), pos(0))
COND2(true, pos(0), neg(s(s(x0)))) → DIFF(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))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QReductionProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(0), neg(s(s(x0)))) → DIFF(pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
COND1(false, pos(0), neg(s(x0))) → COND2(true, 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QReductionProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(0), neg(s(s(x0)))) → DIFF(pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
COND1(false, pos(0), neg(s(x0))) → COND2(true, 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QReductionProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ ForwardInstantiation

                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(0), neg(s(s(x0)))) → DIFF(pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
COND1(false, pos(0), neg(s(x0))) → COND2(true, 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 COND1(false, pos(0), neg(s(x0))) → COND2(true, pos(0), neg(s(x0))) we obtained the following new rules [LPAR04]:

COND1(false, pos(0), neg(s(s(y_0)))) → COND2(true, pos(0), neg(s(s(y_0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QReductionProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ ForwardInstantiation

                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(0), neg(s(s(x0)))) → DIFF(pos(0), neg(s(x0)))
DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0)))
COND1(false, pos(0), neg(s(s(y_0)))) → COND2(true, 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 DIFF(pos(0), neg(s(x0))) → COND1(false, pos(0), neg(s(x0))) we obtained the following new rules [LPAR04]:

DIFF(pos(0), neg(s(s(y_0)))) → COND1(false, pos(0), neg(s(s(y_0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QReductionProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ ForwardInstantiation

                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(0), neg(s(s(x0)))) → DIFF(pos(0), neg(s(x0)))
COND1(false, pos(0), neg(s(s(y_0)))) → COND2(true, pos(0), neg(s(s(y_0))))
DIFF(pos(0), neg(s(s(y_0)))) → COND1(false, 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 COND2(true, pos(0), neg(s(s(x0)))) → DIFF(pos(0), neg(s(x0))) we obtained the following new rules [LPAR04]:

COND2(true, pos(0), neg(s(s(s(y_0))))) → DIFF(pos(0), neg(s(s(y_0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QReductionProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ MRRProof

                                                                                                              ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(0), neg(s(s(y_0)))) → COND2(true, pos(0), neg(s(s(y_0))))
DIFF(pos(0), neg(s(s(y_0)))) → COND1(false, pos(0), neg(s(s(y_0))))
COND2(true, pos(0), neg(s(s(s(y_0))))) → DIFF(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:

COND1(false, pos(0), neg(s(s(y_0)))) → COND2(true, pos(0), neg(s(s(y_0))))
DIFF(pos(0), neg(s(s(y_0)))) → COND1(false, pos(0), neg(s(s(y_0))))
COND2(true, pos(0), neg(s(s(s(y_0))))) → DIFF(pos(0), neg(s(s(y_0))))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(COND2(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(DIFF(x₁, x₂)) = 2 + x₁ + x₂
POL(false) = 0
POL(neg(x₁)) = x₁
POL(pos(x₁)) = x₁
POL(s(x₁)) = 3 + x₁
POL(true) = 0

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                                ↳ UsableRulesProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QReductionProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ UsableRulesProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QReductionProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ MRRProof

                                                                                                                                                      ↳ 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

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                    ↳ QDP

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

DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), minus_nat(s(0), 0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(s(0), s(z1)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(s(0), s(z1)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)

The TRS R consists of the following rules:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), minus_nat(s(0), 0)) at position [1] we obtained the following new rules [LPAR04]:

COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), pos(s(0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                    ↳ QDP

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

DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(s(0), s(z1)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(s(0), s(z1)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), pos(s(0)))

The TRS R consists of the following rules:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(s(0), s(z1))) at position [1] we obtained the following new rules [LPAR04]:

COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(0, z1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                    ↳ QDP

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

DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(s(0), s(z1)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), pos(s(0)))
COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(0, z1))

The TRS R consists of the following rules:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(s(0), s(z1))) at position [1] we obtained the following new rules [LPAR04]:

COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(0, z1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                    ↳ QDP

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

DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), pos(s(0)))
COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(0, z1))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(0, z1))

The TRS R consists of the following rules:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND2(true, y0, pos(x1)) → DIFF(y0, pos(s(x1))) we obtained the following new rules [LPAR04]:

COND2(true, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(0)) → DIFF(pos(s(z0)), pos(s(0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                    ↳ QDP

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

DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
DIFF(neg(0), pos(s(x0))) → COND1(false, neg(0), pos(s(x0)))
COND1(false, neg(0), pos(s(x0))) → COND2(false, neg(0), pos(s(x0)))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(pos(0), pos(s(x0))) → COND1(false, pos(0), pos(s(x0)))
COND1(false, pos(0), pos(s(x0))) → COND2(false, pos(0), pos(s(x0)))
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), pos(s(0)))
COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(0, z1))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(0, z1))
COND2(true, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(0)) → DIFF(pos(s(z0)), pos(s(0)))

The TRS R consists of the following rules:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), pos(s(0)))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(0, z1))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
COND2(true, pos(s(z0)), pos(0)) → DIFF(pos(s(z0)), pos(s(0)))
COND2(true, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(z0)), pos(s(s(z1))))

The TRS R consists of the following rules:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), pos(s(0)))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(0, z1))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
COND2(true, pos(s(z0)), pos(0)) → DIFF(pos(s(z0)), pos(s(0)))
COND2(true, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(z0)), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND2(true, pos(s(z0)), neg(s(z1))) → DIFF(pos(s(z0)), minus_nat(0, z1)) at position [1] we obtained the following new rules [LPAR04]:

COND2(true, pos(s(y0)), neg(s(s(x0)))) → DIFF(pos(s(y0)), neg(s(x0)))
COND2(true, pos(s(y0)), neg(s(0))) → DIFF(pos(s(y0)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), pos(s(0)))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
COND2(true, pos(s(z0)), pos(0)) → DIFF(pos(s(z0)), pos(s(0)))
COND2(true, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(y0)), neg(s(s(x0)))) → DIFF(pos(s(y0)), neg(s(x0)))
COND2(true, pos(s(y0)), neg(s(0))) → DIFF(pos(s(y0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), pos(s(0)))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
COND2(true, pos(s(z0)), pos(0)) → DIFF(pos(s(z0)), pos(s(0)))
COND2(true, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(y0)), neg(s(s(x0)))) → DIFF(pos(s(y0)), neg(s(x0)))
COND2(true, pos(s(y0)), neg(s(0))) → DIFF(pos(s(y0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), pos(s(0)))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1)
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
COND2(true, pos(s(z0)), pos(0)) → DIFF(pos(s(z0)), pos(s(0)))
COND2(true, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(y0)), neg(s(s(x0)))) → DIFF(pos(s(y0)), neg(s(x0)))
COND2(true, pos(s(y0)), neg(s(0))) → DIFF(pos(s(y0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND2(false, pos(x1), y1) → DIFF(pos(s(x1)), y1) we obtained the following new rules [LPAR04]:

COND2(false, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(x0)), neg(0)) → COND2(true, pos(s(x0)), neg(0))
COND2(true, pos(s(z0)), neg(0)) → DIFF(pos(s(z0)), pos(s(0)))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
DIFF(pos(s(x0)), neg(0)) → COND1(false, pos(s(x0)), neg(0))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))
DIFF(pos(s(x0)), pos(0)) → COND1(false, pos(s(x0)), pos(0))
COND1(false, pos(s(x0)), pos(0)) → COND2(true, pos(s(x0)), pos(0))
COND2(true, pos(s(z0)), pos(0)) → DIFF(pos(s(z0)), pos(s(0)))
COND2(true, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(y0)), neg(s(s(x0)))) → DIFF(pos(s(y0)), neg(s(x0)))
COND2(true, pos(s(y0)), neg(s(0))) → DIFF(pos(s(y0)), pos(0))
COND2(false, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 7 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(z0)), pos(s(s(z1))))
DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(false, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule DIFF(pos(s(x0)), pos(s(x1))) → COND1(equal_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))) we obtained the following new rules [LPAR04]:

DIFF(pos(s(z0)), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
DIFF(pos(s(s(z0))), pos(s(z1))) → COND1(equal_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1)))
COND2(true, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(z0)), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
DIFF(pos(s(s(z0))), pos(s(z1))) → COND1(equal_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND1(false, pos(s(x0)), pos(s(x1))) → COND2(greater_int(pos(x0), pos(x1)), pos(s(x0)), pos(s(x1))) we obtained the following new rules [LPAR04]:

COND1(false, pos(s(s(z0))), pos(s(z1))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(z0)), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
DIFF(pos(s(s(z0))), pos(s(z1))) → COND1(equal_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(s(z0))), pos(s(z1))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND2(true, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(z0)), pos(s(s(z1)))) we obtained the following new rules [LPAR04]:

COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND2(false, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
DIFF(pos(s(s(z0))), pos(s(z1))) → COND1(equal_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(s(z0))), pos(s(z1))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND2(false, pos(s(z0)), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(z1))) we obtained the following new rules [LPAR04]:

COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

DIFF(pos(s(z0)), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
DIFF(pos(s(s(z0))), pos(s(z1))) → COND1(equal_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(s(z0))), pos(s(z1))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule DIFF(pos(s(z0)), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1)))) we obtained the following new rules [LPAR04]:

DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(s(z0))), pos(s(x1))), pos(s(s(s(z0)))), pos(s(s(x1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

DIFF(pos(s(s(z0))), pos(s(z1))) → COND1(equal_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(s(z0))), pos(s(z1))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(s(z0))), pos(s(x1))), pos(s(s(s(z0)))), pos(s(s(x1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1)))) at position [0] we obtained the following new rules [LPAR04]:

DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

DIFF(pos(s(s(z0))), pos(s(z1))) → COND1(equal_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(s(z0))), pos(s(z1))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(s(z0))), pos(s(x1))), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(s(z0))), pos(s(x1))), pos(s(s(s(z0)))), pos(s(s(x1)))) at position [0] we obtained the following new rules [LPAR04]:

DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

DIFF(pos(s(s(z0))), pos(s(z1))) → COND1(equal_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(s(z0))), pos(s(z1))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule DIFF(pos(s(s(z0))), pos(s(z1))) → COND1(equal_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1))) we obtained the following new rules [LPAR04]:

DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(s(x0)), pos(s(s(z1)))), pos(s(s(x0))), pos(s(s(s(z1)))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(s(z0))), pos(s(z1))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(s(x0)), pos(s(s(z1)))), pos(s(s(x0))), pos(s(s(s(z1)))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(s(z0))), pos(s(z1))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(s(x0)), pos(s(s(z1)))), pos(s(s(x0))), pos(s(s(s(z1)))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(s(x0)), pos(s(s(z1)))), pos(s(s(x0))), pos(s(s(s(z1))))) at position [0] we obtained the following new rules [LPAR04]:

DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(s(z0))), pos(s(z1))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1)))
COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND1(false, pos(s(s(z0))), pos(s(z1))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(z0))), pos(s(z1))) we obtained the following new rules [LPAR04]:

COND1(false, pos(s(s(s(z0)))), pos(s(z1))) → COND2(greater_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
COND1(false, pos(s(s(x0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(s(x0)), pos(s(s(z1)))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(s(z0))), pos(s(z1))), pos(s(s(s(z0)))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(z1))) → COND2(greater_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
COND1(false, pos(s(s(x0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(s(x0)), pos(s(s(z1)))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(s(z0))), pos(s(z1))), pos(s(s(s(z0)))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND1(false, pos(s(s(x0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(s(x0)), pos(s(s(z1)))), pos(s(s(x0))), pos(s(s(s(z1))))) at position [0] we obtained the following new rules [LPAR04]:

COND1(false, pos(s(s(x0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(z1))) → COND2(greater_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(s(z0))), pos(s(z1))), pos(s(s(s(z0)))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1))))
COND1(false, pos(s(s(x0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(s(z0))), pos(s(z1))), pos(s(s(s(z0)))), pos(s(s(z1)))) at position [0] we obtained the following new rules [LPAR04]:

COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(s(z0)))), pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(z1))) → COND2(greater_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1))))
COND1(false, pos(s(s(x0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(s(z0)))), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1)))) at position [0] we obtained the following new rules [LPAR04]:

COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1))))
COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(z1))) → COND2(greater_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
COND1(false, pos(s(s(x0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(s(z0)))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND1(false, pos(s(z0)), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(z0)), pos(s(s(z1)))) we obtained the following new rules [LPAR04]:

COND1(false, pos(s(s(z0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(s(z0)), pos(s(s(z1)))), pos(s(s(z0))), pos(s(s(s(z1)))))
COND1(false, pos(s(z0)), pos(s(s(s(z1))))) → COND2(greater_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(s(z0))), pos(s(z1))), pos(s(s(s(z0)))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(z1))) → COND2(greater_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
COND1(false, pos(s(s(x0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(s(z0)))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(s(z0)), pos(s(s(z1)))), pos(s(s(z0))), pos(s(s(s(z1)))))
COND1(false, pos(s(z0)), pos(s(s(s(z1))))) → COND2(greater_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(s(z0))), pos(s(z1))), pos(s(s(s(z0)))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND1(false, pos(s(s(z0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(s(z0)), pos(s(s(z1)))), pos(s(s(z0))), pos(s(s(s(z1))))) at position [0] we obtained the following new rules [LPAR04]:

COND1(false, pos(s(s(z0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(z0), pos(s(z1))), pos(s(s(z0))), pos(s(s(s(z1)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(z1))) → COND2(greater_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
COND1(false, pos(s(s(x0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(s(z0)))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
COND1(false, pos(s(z0)), pos(s(s(s(z1))))) → COND2(greater_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(s(z0))), pos(s(z1))), pos(s(s(s(z0)))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(s(z0)))), pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(z1))) → COND2(greater_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
COND1(false, pos(s(s(x0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(s(z0)))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
COND1(false, pos(s(z0)), pos(s(s(s(z1))))) → COND2(greater_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(s(z1))), pos(s(s(z0))), pos(s(s(z1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(z1))) → COND2(greater_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
COND1(false, pos(s(s(x0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(s(z0)))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
COND1(false, pos(s(z0)), pos(s(s(s(z1))))) → COND2(greater_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ MNOCProof

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(z1))) → COND2(greater_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
COND1(false, pos(s(s(x0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(s(z0)))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
COND1(false, pos(s(z0)), pos(s(s(s(z1))))) → COND2(greater_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

equal_int(pos(0), pos(0))
equal_int(pos(0), pos(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), pos(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), pos(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Rewriting

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ Instantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ Instantiation

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Rewriting

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ Rewriting

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ Rewriting

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ MNOCProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                              ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(z0)), pos(s(s(s(z1)))))
COND2(true, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(z0)), pos(s(s(z1)))) → DIFF(pos(s(s(z0))), pos(s(s(z1))))
COND2(false, pos(s(s(z0))), pos(s(z1))) → DIFF(pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(z0)), pos(s(s(s(z1))))) → COND1(equal_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))
DIFF(pos(s(s(z0))), pos(s(s(z1)))) → COND1(equal_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
DIFF(pos(s(s(s(z0)))), pos(s(s(x1)))) → COND1(equal_int(pos(s(z0)), pos(x1)), pos(s(s(s(z0)))), pos(s(s(x1))))
DIFF(pos(s(s(s(z0)))), pos(s(z1))) → COND1(equal_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
DIFF(pos(s(s(x0))), pos(s(s(s(z1))))) → COND1(equal_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(z1))) → COND2(greater_int(pos(s(s(z0))), pos(z1)), pos(s(s(s(z0)))), pos(s(z1)))
COND1(false, pos(s(s(x0))), pos(s(s(s(z1))))) → COND2(greater_int(pos(x0), pos(s(z1))), pos(s(s(x0))), pos(s(s(s(z1)))))
COND1(false, pos(s(s(s(z0)))), pos(s(s(z1)))) → COND2(greater_int(pos(s(z0)), pos(z1)), pos(s(s(s(z0)))), pos(s(s(z1))))
COND1(false, pos(s(s(z0))), pos(s(s(z1)))) → COND2(greater_int(pos(z0), pos(z1)), pos(s(s(z0))), pos(s(s(z1))))
COND1(false, pos(s(z0)), pos(s(s(s(z1))))) → COND2(greater_int(pos(z0), pos(s(s(z1)))), pos(s(z0)), pos(s(s(s(z1)))))

The TRS R consists of the following rules:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(s(y0)), neg(s(s(x0)))) → DIFF(pos(s(y0)), neg(s(x0)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(s(y0)), neg(s(s(x0)))) → DIFF(pos(s(y0)), neg(s(x0)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))

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

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(s(y0)), neg(s(s(x0)))) → DIFF(pos(s(y0)), neg(s(x0)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1)))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND1(false, pos(s(x0)), neg(s(x1))) → COND2(true, pos(s(x0)), neg(s(x1))) we obtained the following new rules [LPAR04]:

COND1(false, pos(s(x0)), neg(s(s(y_1)))) → COND2(true, pos(s(x0)), neg(s(s(y_1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(s(y0)), neg(s(s(x0)))) → DIFF(pos(s(y0)), neg(s(x0)))
DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1)))
COND1(false, pos(s(x0)), neg(s(s(y_1)))) → COND2(true, pos(s(x0)), neg(s(s(y_1))))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule DIFF(pos(s(x0)), neg(s(x1))) → COND1(false, pos(s(x0)), neg(s(x1))) we obtained the following new rules [LPAR04]:

DIFF(pos(s(x0)), neg(s(s(y_1)))) → COND1(false, pos(s(x0)), neg(s(s(y_1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND2(true, pos(s(y0)), neg(s(s(x0)))) → DIFF(pos(s(y0)), neg(s(x0)))
COND1(false, pos(s(x0)), neg(s(s(y_1)))) → COND2(true, pos(s(x0)), neg(s(s(y_1))))
DIFF(pos(s(x0)), neg(s(s(y_1)))) → COND1(false, pos(s(x0)), neg(s(s(y_1))))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND2(true, pos(s(y0)), neg(s(s(x0)))) → DIFF(pos(s(y0)), neg(s(x0))) we obtained the following new rules [LPAR04]:

COND2(true, pos(s(x0)), neg(s(s(s(y_1))))) → DIFF(pos(s(x0)), neg(s(s(y_1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ MRRProof

                                                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, pos(s(x0)), neg(s(s(y_1)))) → COND2(true, pos(s(x0)), neg(s(s(y_1))))
DIFF(pos(s(x0)), neg(s(s(y_1)))) → COND1(false, pos(s(x0)), neg(s(s(y_1))))
COND2(true, pos(s(x0)), neg(s(s(s(y_1))))) → DIFF(pos(s(x0)), neg(s(s(y_1))))

Used ordering: Polynomial interpretation [POLO]:

POL(COND1(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(COND2(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(DIFF(x₁, x₂)) = 2 + x₁ + x₂
POL(false) = 0
POL(neg(x₁)) = x₁
POL(pos(x₁)) = x₁
POL(s(x₁)) = 3 + x₁
POL(true) = 0

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Narrowing

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ UsableRulesProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QReductionProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Instantiation

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ AND

                                                                                                                                                              ↳ QDP

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QReductionProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ MRRProof

                                                                                                                                                                                      ↳ 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

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                    ↳ QDP

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

DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(0, z1))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))

The TRS R consists of the following rules:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                    ↳ QDP

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

DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1)
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(0, z1))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND2(false, neg(x1), y1) → DIFF(minus_nat(s(0), x1), y1) we obtained the following new rules [LPAR04]:

COND2(false, neg(s(z0)), pos(0)) → DIFF(minus_nat(s(0), s(z0)), pos(0))
COND2(false, neg(s(z0)), pos(s(z1))) → DIFF(minus_nat(s(0), s(z0)), pos(s(z1)))
COND2(false, neg(s(z0)), neg(0)) → DIFF(minus_nat(s(0), s(z0)), neg(0))
COND2(false, neg(s(z0)), neg(s(z1))) → DIFF(minus_nat(s(0), s(z0)), neg(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                    ↳ QDP

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

DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(0, z1))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
COND2(false, neg(s(z0)), pos(0)) → DIFF(minus_nat(s(0), s(z0)), pos(0))
COND2(false, neg(s(z0)), pos(s(z1))) → DIFF(minus_nat(s(0), s(z0)), pos(s(z1)))
COND2(false, neg(s(z0)), neg(0)) → DIFF(minus_nat(s(0), s(z0)), neg(0))
COND2(false, neg(s(z0)), neg(s(z1))) → DIFF(minus_nat(s(0), s(z0)), neg(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
COND2(false, neg(s(z0)), neg(0)) → DIFF(minus_nat(s(0), s(z0)), neg(0))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
COND2(false, neg(s(z0)), neg(0)) → DIFF(minus_nat(s(0), s(z0)), neg(0))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
COND2(false, neg(s(z0)), neg(0)) → DIFF(minus_nat(s(0), s(z0)), neg(0))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))

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 COND2(false, neg(s(z0)), neg(0)) → DIFF(minus_nat(s(0), s(z0)), neg(0)) at position [0] we obtained the following new rules [LPAR04]:

COND2(false, neg(s(z0)), neg(0)) → DIFF(minus_nat(0, z0), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND2(false, neg(s(z0)), neg(0)) → DIFF(minus_nat(0, z0), neg(0))

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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND2(false, neg(s(z0)), neg(0)) → DIFF(minus_nat(0, z0), neg(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.
By narrowing [LPAR04] the rule COND2(false, neg(s(z0)), neg(0)) → DIFF(minus_nat(0, z0), neg(0)) at position [0] we obtained the following new rules [LPAR04]:

COND2(false, neg(s(0)), neg(0)) → DIFF(pos(0), neg(0))
COND2(false, neg(s(s(x0))), neg(0)) → DIFF(neg(s(x0)), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND2(false, neg(s(0)), neg(0)) → DIFF(pos(0), neg(0))
COND2(false, neg(s(s(x0))), neg(0)) → DIFF(neg(s(x0)), neg(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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), neg(0)) → DIFF(neg(s(x0)), neg(0))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(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))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), neg(0)) → DIFF(neg(s(x0)), neg(0))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))

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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), neg(0)) → DIFF(neg(s(x0)), neg(0))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND1(false, neg(s(x0)), neg(0)) → COND2(false, neg(s(x0)), neg(0)) we obtained the following new rules [LPAR04]:

COND1(false, neg(s(s(y_0))), neg(0)) → COND2(false, neg(s(s(y_0))), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), neg(0)) → DIFF(neg(s(x0)), neg(0))
DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0))
COND1(false, neg(s(s(y_0))), neg(0)) → COND2(false, neg(s(s(y_0))), neg(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule DIFF(neg(s(x0)), neg(0)) → COND1(false, neg(s(x0)), neg(0)) we obtained the following new rules [LPAR04]:

DIFF(neg(s(s(y_0))), neg(0)) → COND1(false, neg(s(s(y_0))), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), neg(0)) → DIFF(neg(s(x0)), neg(0))
COND1(false, neg(s(s(y_0))), neg(0)) → COND2(false, neg(s(s(y_0))), neg(0))
DIFF(neg(s(s(y_0))), neg(0)) → COND1(false, neg(s(s(y_0))), neg(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND2(false, neg(s(s(x0))), neg(0)) → DIFF(neg(s(x0)), neg(0)) we obtained the following new rules [LPAR04]:

COND2(false, neg(s(s(s(y_0)))), neg(0)) → DIFF(neg(s(s(y_0))), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ MRRProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(s(y_0))), neg(0)) → COND2(false, neg(s(s(y_0))), neg(0))
DIFF(neg(s(s(y_0))), neg(0)) → COND1(false, neg(s(s(y_0))), neg(0))
COND2(false, neg(s(s(s(y_0)))), neg(0)) → DIFF(neg(s(s(y_0))), neg(0))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(COND2(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(DIFF(x₁, x₂)) = 2 + x₁ + x₂
POL(false) = 0
POL(neg(x₁)) = x₁
POL(s(x₁)) = 3 + x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ MRRProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ PisEmptyProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
COND2(false, neg(s(z0)), pos(s(z1))) → DIFF(minus_nat(s(0), s(z0)), pos(s(z1)))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
COND2(false, neg(s(z0)), pos(s(z1))) → DIFF(minus_nat(s(0), s(z0)), pos(s(z1)))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(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))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
COND2(false, neg(s(z0)), pos(s(z1))) → DIFF(minus_nat(s(0), s(z0)), pos(s(z1)))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(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 COND2(false, neg(s(z0)), pos(s(z1))) → DIFF(minus_nat(s(0), s(z0)), pos(s(z1))) at position [0] we obtained the following new rules [LPAR04]:

COND2(false, neg(s(z0)), pos(s(z1))) → DIFF(minus_nat(0, z0), pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND2(false, neg(s(z0)), pos(s(z1))) → DIFF(minus_nat(0, z0), pos(s(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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND2(false, neg(s(z0)), pos(s(z1))) → DIFF(minus_nat(0, z0), pos(s(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 COND2(false, neg(s(z0)), pos(s(z1))) → DIFF(minus_nat(0, z0), pos(s(z1))) at position [0] we obtained the following new rules [LPAR04]:

COND2(false, neg(s(s(x0))), pos(s(y1))) → DIFF(neg(s(x0)), pos(s(y1)))
COND2(false, neg(s(0)), pos(s(y1))) → DIFF(pos(0), pos(s(y1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND2(false, neg(s(s(x0))), pos(s(y1))) → DIFF(neg(s(x0)), pos(s(y1)))
COND2(false, neg(s(0)), pos(s(y1))) → DIFF(pos(0), pos(s(y1)))

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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), pos(s(y1))) → DIFF(neg(s(x0)), pos(s(y1)))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), pos(s(y1))) → DIFF(neg(s(x0)), pos(s(y1)))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), pos(s(y1))) → DIFF(neg(s(x0)), pos(s(y1)))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1)))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND1(false, neg(s(x0)), pos(s(x1))) → COND2(false, neg(s(x0)), pos(s(x1))) we obtained the following new rules [LPAR04]:

COND1(false, neg(s(s(y_0))), pos(s(x1))) → COND2(false, neg(s(s(y_0))), pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), pos(s(y1))) → DIFF(neg(s(x0)), pos(s(y1)))
DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1)))
COND1(false, neg(s(s(y_0))), pos(s(x1))) → COND2(false, neg(s(s(y_0))), pos(s(x1)))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule DIFF(neg(s(x0)), pos(s(x1))) → COND1(false, neg(s(x0)), pos(s(x1))) we obtained the following new rules [LPAR04]:

DIFF(neg(s(s(y_0))), pos(s(x1))) → COND1(false, neg(s(s(y_0))), pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), pos(s(y1))) → DIFF(neg(s(x0)), pos(s(y1)))
COND1(false, neg(s(s(y_0))), pos(s(x1))) → COND2(false, neg(s(s(y_0))), pos(s(x1)))
DIFF(neg(s(s(y_0))), pos(s(x1))) → COND1(false, neg(s(s(y_0))), pos(s(x1)))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND2(false, neg(s(s(x0))), pos(s(y1))) → DIFF(neg(s(x0)), pos(s(y1))) we obtained the following new rules [LPAR04]:

COND2(false, neg(s(s(s(y_0)))), pos(s(x1))) → DIFF(neg(s(s(y_0))), pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ MRRProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(s(y_0))), pos(s(x1))) → COND2(false, neg(s(s(y_0))), pos(s(x1)))
DIFF(neg(s(s(y_0))), pos(s(x1))) → COND1(false, neg(s(s(y_0))), pos(s(x1)))
COND2(false, neg(s(s(s(y_0)))), pos(s(x1))) → DIFF(neg(s(s(y_0))), pos(s(x1)))

Used ordering: Polynomial interpretation [POLO]:

POL(COND1(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(COND2(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(DIFF(x₁, x₂)) = 2 + x₁ + x₂
POL(false) = 0
POL(neg(x₁)) = x₁
POL(pos(x₁)) = x₁
POL(s(x₁)) = 3 + x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ MRRProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ PisEmptyProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
COND2(false, neg(s(z0)), pos(0)) → DIFF(minus_nat(s(0), s(z0)), pos(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
COND2(false, neg(s(z0)), pos(0)) → DIFF(minus_nat(s(0), s(z0)), pos(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
COND2(false, neg(s(z0)), pos(0)) → DIFF(minus_nat(s(0), s(z0)), pos(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))

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 COND2(false, neg(s(z0)), pos(0)) → DIFF(minus_nat(s(0), s(z0)), pos(0)) at position [0] we obtained the following new rules [LPAR04]:

COND2(false, neg(s(z0)), pos(0)) → DIFF(minus_nat(0, z0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND2(false, neg(s(z0)), pos(0)) → DIFF(minus_nat(0, z0), pos(0))

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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND2(false, neg(s(z0)), pos(0)) → DIFF(minus_nat(0, z0), 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.
By narrowing [LPAR04] the rule COND2(false, neg(s(z0)), pos(0)) → DIFF(minus_nat(0, z0), pos(0)) at position [0] we obtained the following new rules [LPAR04]:

COND2(false, neg(s(0)), pos(0)) → DIFF(pos(0), pos(0))
COND2(false, neg(s(s(x0))), pos(0)) → DIFF(neg(s(x0)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND2(false, neg(s(0)), pos(0)) → DIFF(pos(0), pos(0))
COND2(false, neg(s(s(x0))), pos(0)) → DIFF(neg(s(x0)), 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), pos(0)) → DIFF(neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), 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))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QReductionProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Rewriting

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), pos(0)) → DIFF(neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))

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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), pos(0)) → DIFF(neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND1(false, neg(s(x0)), pos(0)) → COND2(false, neg(s(x0)), pos(0)) we obtained the following new rules [LPAR04]:

COND1(false, neg(s(s(y_0))), pos(0)) → COND2(false, neg(s(s(y_0))), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), pos(0)) → DIFF(neg(s(x0)), pos(0))
DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0))
COND1(false, neg(s(s(y_0))), pos(0)) → COND2(false, neg(s(s(y_0))), pos(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule DIFF(neg(s(x0)), pos(0)) → COND1(false, neg(s(x0)), pos(0)) we obtained the following new rules [LPAR04]:

DIFF(neg(s(s(y_0))), pos(0)) → COND1(false, neg(s(s(y_0))), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), pos(0)) → DIFF(neg(s(x0)), pos(0))
COND1(false, neg(s(s(y_0))), pos(0)) → COND2(false, neg(s(s(y_0))), pos(0))
DIFF(neg(s(s(y_0))), pos(0)) → COND1(false, neg(s(s(y_0))), pos(0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND2(false, neg(s(s(x0))), pos(0)) → DIFF(neg(s(x0)), pos(0)) we obtained the following new rules [LPAR04]:

COND2(false, neg(s(s(s(y_0)))), pos(0)) → DIFF(neg(s(s(y_0))), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ MRRProof

                                                                                                                                                  ↳ QDP

                                                                                                    ↳ QDP

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

COND1(false, neg(s(s(y_0))), pos(0)) → COND2(false, neg(s(s(y_0))), pos(0))
DIFF(neg(s(s(y_0))), pos(0)) → COND1(false, neg(s(s(y_0))), pos(0))
COND2(false, neg(s(s(s(y_0)))), pos(0)) → DIFF(neg(s(s(y_0))), pos(0))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(COND2(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(DIFF(x₁, x₂)) = 2 + x₁ + x₂
POL(false) = 0
POL(neg(x₁)) = x₁
POL(pos(x₁)) = x₁
POL(s(x₁)) = 3 + x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ 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

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ MRRProof

                                                                                                                                                                                                  ↳ 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

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                    ↳ QDP

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

DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(0, z1))
COND2(false, neg(s(z0)), neg(s(z1))) → DIFF(minus_nat(s(0), s(z0)), neg(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND2(false, neg(s(z0)), neg(s(z1))) → DIFF(minus_nat(s(0), s(z0)), neg(s(z1))) at position [0] we obtained the following new rules [LPAR04]:

COND2(false, neg(s(z0)), neg(s(z1))) → DIFF(minus_nat(0, z0), neg(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                    ↳ QDP

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

DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(0, z1))
COND2(false, neg(s(z0)), neg(s(z1))) → DIFF(minus_nat(0, z0), neg(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                    ↳ QDP

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

DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(0, z1))
COND2(false, neg(s(z0)), neg(s(z1))) → DIFF(minus_nat(0, z0), neg(s(z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND2(true, neg(s(z0)), neg(s(z1))) → DIFF(neg(s(z0)), minus_nat(0, z1)) at position [1] we obtained the following new rules [LPAR04]:

COND2(true, neg(s(y0)), neg(s(0))) → DIFF(neg(s(y0)), pos(0))
COND2(true, neg(s(y0)), neg(s(s(x0)))) → DIFF(neg(s(y0)), neg(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                    ↳ QDP

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

DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(false, neg(s(z0)), neg(s(z1))) → DIFF(minus_nat(0, z0), neg(s(z1)))
COND2(true, neg(s(y0)), neg(s(0))) → DIFF(neg(s(y0)), pos(0))
COND2(true, neg(s(y0)), neg(s(s(x0)))) → DIFF(neg(s(y0)), 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))
equal_int(neg(0), neg(0)) → true
equal_int(neg(0), neg(s(y))) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(false, neg(s(z0)), neg(s(z1))) → DIFF(minus_nat(0, z0), neg(s(z1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(true, neg(s(y0)), neg(s(s(x0)))) → DIFF(neg(s(y0)), 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))
equal_int(neg(0), neg(0)) → true
equal_int(neg(0), neg(s(y))) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND2(false, neg(s(z0)), neg(s(z1))) → DIFF(minus_nat(0, z0), neg(s(z1))) at position [0] we obtained the following new rules [LPAR04]:

COND2(false, neg(s(s(x0))), neg(s(y1))) → DIFF(neg(s(x0)), neg(s(y1)))
COND2(false, neg(s(0)), neg(s(y1))) → DIFF(pos(0), neg(s(y1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                    ↳ QDP

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

COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(true, neg(s(y0)), neg(s(s(x0)))) → DIFF(neg(s(y0)), neg(s(x0)))
COND2(false, neg(s(s(x0))), neg(s(y1))) → DIFF(neg(s(x0)), neg(s(y1)))
COND2(false, neg(s(0)), neg(s(y1))) → DIFF(pos(0), neg(s(y1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                    ↳ QDP

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

COND2(true, neg(s(y0)), neg(s(s(x0)))) → DIFF(neg(s(y0)), neg(s(x0)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(false, neg(s(s(x0))), neg(s(y1))) → DIFF(neg(s(x0)), neg(s(y1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                    ↳ QDP

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

COND2(true, neg(s(y0)), neg(s(s(x0)))) → DIFF(neg(s(y0)), neg(s(x0)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(false, neg(s(s(x0))), neg(s(y1))) → DIFF(neg(s(x0)), neg(s(y1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                    ↳ QDP

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

COND2(true, neg(s(y0)), neg(s(s(x0)))) → DIFF(neg(s(y0)), neg(s(x0)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(false, neg(s(s(x0))), neg(s(y1))) → DIFF(neg(s(x0)), neg(s(y1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND1(false, neg(s(x0)), neg(s(x1))) → COND2(greater_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))) we obtained the following new rules [LPAR04]:

COND1(false, neg(s(s(y_1))), neg(s(x1))) → COND2(greater_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(s(y_2)))) → COND2(greater_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                    ↳ QDP

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

COND2(true, neg(s(y0)), neg(s(s(x0)))) → DIFF(neg(s(y0)), neg(s(x0)))
DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1)))
COND2(false, neg(s(s(x0))), neg(s(y1))) → DIFF(neg(s(x0)), neg(s(y1)))
COND1(false, neg(s(s(y_1))), neg(s(x1))) → COND2(greater_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(s(y_2)))) → COND2(greater_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule DIFF(neg(s(x0)), neg(s(x1))) → COND1(equal_int(neg(x0), neg(x1)), neg(s(x0)), neg(s(x1))) we obtained the following new rules [LPAR04]:

DIFF(neg(s(s(y_1))), neg(s(x1))) → COND1(equal_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(s(y_2)))) → COND1(equal_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                    ↳ QDP

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

COND2(true, neg(s(y0)), neg(s(s(x0)))) → DIFF(neg(s(y0)), neg(s(x0)))
COND2(false, neg(s(s(x0))), neg(s(y1))) → DIFF(neg(s(x0)), neg(s(y1)))
COND1(false, neg(s(s(y_1))), neg(s(x1))) → COND2(greater_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(s(y_2)))) → COND2(greater_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))
DIFF(neg(s(s(y_1))), neg(s(x1))) → COND1(equal_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(s(y_2)))) → COND1(equal_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND2(true, neg(s(y0)), neg(s(s(x0)))) → DIFF(neg(s(y0)), neg(s(x0))) we obtained the following new rules [LPAR04]:

COND2(true, neg(s(s(y_0))), neg(s(s(x1)))) → DIFF(neg(s(s(y_0))), neg(s(x1)))
COND2(true, neg(s(x0)), neg(s(s(s(y_1))))) → DIFF(neg(s(x0)), neg(s(s(y_1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ ForwardInstantiation

                                                                                                    ↳ QDP

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

COND2(false, neg(s(s(x0))), neg(s(y1))) → DIFF(neg(s(x0)), neg(s(y1)))
COND1(false, neg(s(s(y_1))), neg(s(x1))) → COND2(greater_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(s(y_2)))) → COND2(greater_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))
DIFF(neg(s(s(y_1))), neg(s(x1))) → COND1(equal_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(s(y_2)))) → COND1(equal_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))
COND2(true, neg(s(s(y_0))), neg(s(s(x1)))) → DIFF(neg(s(s(y_0))), neg(s(x1)))
COND2(true, neg(s(x0)), neg(s(s(s(y_1))))) → DIFF(neg(s(x0)), neg(s(s(y_1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule COND2(false, neg(s(s(x0))), neg(s(y1))) → DIFF(neg(s(x0)), neg(s(y1))) we obtained the following new rules [LPAR04]:

COND2(false, neg(s(s(x0))), neg(s(s(y_1)))) → DIFF(neg(s(x0)), neg(s(s(y_1))))
COND2(false, neg(s(s(s(y_0)))), neg(s(x1))) → DIFF(neg(s(s(y_0))), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                    ↳ QDP

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

COND1(false, neg(s(s(y_1))), neg(s(x1))) → COND2(greater_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(s(y_2)))) → COND2(greater_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))
DIFF(neg(s(s(y_1))), neg(s(x1))) → COND1(equal_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(s(y_2)))) → COND1(equal_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))
COND2(true, neg(s(s(y_0))), neg(s(s(x1)))) → DIFF(neg(s(s(y_0))), neg(s(x1)))
COND2(true, neg(s(x0)), neg(s(s(s(y_1))))) → DIFF(neg(s(x0)), neg(s(s(y_1))))
COND2(false, neg(s(s(x0))), neg(s(s(y_1)))) → DIFF(neg(s(x0)), neg(s(s(y_1))))
COND2(false, neg(s(s(s(y_0)))), neg(s(x1))) → DIFF(neg(s(s(y_0))), neg(s(x1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

The following pairs can be oriented strictly and are deleted.

COND2(false, neg(s(s(x0))), neg(s(s(y_1)))) → DIFF(neg(s(x0)), neg(s(s(y_1))))
COND2(false, neg(s(s(s(y_0)))), neg(s(x1))) → DIFF(neg(s(s(y_0))), neg(s(x1)))

The remaining pairs can at least be oriented weakly.

COND1(false, neg(s(s(y_1))), neg(s(x1))) → COND2(greater_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(s(y_2)))) → COND2(greater_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))
DIFF(neg(s(s(y_1))), neg(s(x1))) → COND1(equal_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(s(y_2)))) → COND1(equal_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))
COND2(true, neg(s(s(y_0))), neg(s(s(x1)))) → DIFF(neg(s(s(y_0))), neg(s(x1)))
COND2(true, neg(s(x0)), neg(s(s(s(y_1))))) → DIFF(neg(s(x0)), neg(s(s(y_1))))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = x₂
POL(COND2(x₁, x₂, x₃)) = x₂
POL(DIFF(x₁, x₂)) = x₁
POL(equal_int(x₁, x₂)) = 0
POL(false) = 0
POL(greater_int(x₁, x₂)) = 0
POL(neg(x₁)) = 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                    ↳ QDP

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

COND1(false, neg(s(s(y_1))), neg(s(x1))) → COND2(greater_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(s(y_2)))) → COND2(greater_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))
DIFF(neg(s(s(y_1))), neg(s(x1))) → COND1(equal_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(s(y_2)))) → COND1(equal_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))
COND2(true, neg(s(s(y_0))), neg(s(s(x1)))) → DIFF(neg(s(s(y_0))), neg(s(x1)))
COND2(true, neg(s(x0)), neg(s(s(s(y_1))))) → DIFF(neg(s(x0)), neg(s(s(y_1))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

The following pairs can be oriented strictly and are deleted.

COND2(true, neg(s(s(y_0))), neg(s(s(x1)))) → DIFF(neg(s(s(y_0))), neg(s(x1)))
COND2(true, neg(s(x0)), neg(s(s(s(y_1))))) → DIFF(neg(s(x0)), neg(s(s(y_1))))

The remaining pairs can at least be oriented weakly.

COND1(false, neg(s(s(y_1))), neg(s(x1))) → COND2(greater_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(s(y_2)))) → COND2(greater_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))
DIFF(neg(s(s(y_1))), neg(s(x1))) → COND1(equal_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(s(y_2)))) → COND1(equal_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = x₃
POL(COND2(x₁, x₂, x₃)) = x₃
POL(DIFF(x₁, x₂)) = x₂
POL(equal_int(x₁, x₂)) = 0
POL(false) = 0
POL(greater_int(x₁, x₂)) = 0
POL(neg(x₁)) = 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                      ↳ Rewriting

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ AND

                                                                                                              ↳ QDP

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Rewriting

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Instantiation

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ AND

                                                                                                                                    ↳ QDP

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ UsableRulesProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                ↳ AND

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Rewriting

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ UsableRulesProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Narrowing

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QReductionProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ ForwardInstantiation

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ ForwardInstantiation

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                    ↳ QDP

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

COND1(false, neg(s(s(y_1))), neg(s(x1))) → COND2(greater_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
COND1(false, neg(s(x0)), neg(s(s(y_2)))) → COND2(greater_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))
DIFF(neg(s(s(y_1))), neg(s(x1))) → COND1(equal_int(neg(s(y_1)), neg(x1)), neg(s(s(y_1))), neg(s(x1)))
DIFF(neg(s(x0)), neg(s(s(y_2)))) → COND1(equal_int(neg(x0), neg(s(y_2))), neg(s(x0)), neg(s(s(y_2))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

                                                                                                      ↳ UsableRulesProof

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

DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
COND2(true, neg(0), neg(s(z0))) → DIFF(neg(0), minus_nat(s(0), s(z0)))

The TRS R consists of the following rules:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

                                                                                                      ↳ UsableRulesProof

                                                                                                        ↳ QDP

                                                                                                          ↳ QReductionProof

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

DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
COND2(true, neg(0), neg(s(z0))) → DIFF(neg(0), minus_nat(s(0), s(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))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

                                                                                                      ↳ UsableRulesProof

                                                                                                        ↳ QDP

                                                                                                          ↳ QReductionProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

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

DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
COND2(true, neg(0), neg(s(z0))) → DIFF(neg(0), minus_nat(s(0), s(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))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND2(true, neg(0), neg(s(z0))) → DIFF(neg(0), minus_nat(s(0), s(z0))) at position [1] we obtained the following new rules [LPAR04]:

COND2(true, neg(0), neg(s(z0))) → DIFF(neg(0), minus_nat(0, z0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

                                                                                                      ↳ UsableRulesProof

                                                                                                        ↳ QDP

                                                                                                          ↳ QReductionProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                                                ↳ QDP

                                                                                                                  ↳ UsableRulesProof

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

DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
COND2(true, neg(0), neg(s(z0))) → DIFF(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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

                                                                                                      ↳ UsableRulesProof

                                                                                                        ↳ QDP

                                                                                                          ↳ QReductionProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                                                ↳ QDP

                                                                                                                  ↳ UsableRulesProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

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

DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
COND2(true, neg(0), neg(s(z0))) → DIFF(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 COND2(true, neg(0), neg(s(z0))) → DIFF(neg(0), minus_nat(0, z0)) at position [1] we obtained the following new rules [LPAR04]:

COND2(true, neg(0), neg(s(s(x0)))) → DIFF(neg(0), neg(s(x0)))
COND2(true, neg(0), neg(s(0))) → DIFF(neg(0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ QDP

                                                                                                    ↳ QDP

                                                                                                      ↳ UsableRulesProof

                                                                                                        ↳ QDP

                                                                                                          ↳ QReductionProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                                                ↳ QDP

                                                                                                                  ↳ UsableRulesProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

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

DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
COND2(true, neg(0), neg(s(s(x0)))) → DIFF(neg(0), neg(s(x0)))
COND2(true, neg(0), neg(s(0))) → DIFF(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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ UsableRulesProof

                                                                                      ↳ QDP

                                                                                        ↳ QReductionProof

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ AND

                                                                                                    ↳ 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:

COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
COND2(true, neg(0), neg(s(s(x0)))) → DIFF(neg(0), neg(s(x0)))
DIFF(neg(0), neg(s(x0))) → COND1(false, 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ 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

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

COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
COND2(true, neg(0), neg(s(s(x0)))) → DIFF(neg(0), neg(s(x0)))
DIFF(neg(0), neg(s(x0))) → COND1(false, 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ 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

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

COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0)))
COND2(true, neg(0), neg(s(s(x0)))) → DIFF(neg(0), neg(s(x0)))
DIFF(neg(0), neg(s(x0))) → COND1(false, 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 COND1(false, neg(0), neg(s(x0))) → COND2(true, neg(0), neg(s(x0))) we obtained the following new rules [LPAR04]:

COND1(false, neg(0), neg(s(s(y_0)))) → COND2(true, neg(0), neg(s(s(y_0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ 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

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

COND2(true, neg(0), neg(s(s(x0)))) → DIFF(neg(0), neg(s(x0)))
DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(s(y_0)))) → COND2(true, 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 DIFF(neg(0), neg(s(x0))) → COND1(false, neg(0), neg(s(x0))) we obtained the following new rules [LPAR04]:

DIFF(neg(0), neg(s(s(y_0)))) → COND1(false, neg(0), neg(s(s(y_0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ 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

                                                                                                                                              ↳ ForwardInstantiation

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

COND2(true, neg(0), neg(s(s(x0)))) → DIFF(neg(0), neg(s(x0)))
COND1(false, neg(0), neg(s(s(y_0)))) → COND2(true, neg(0), neg(s(s(y_0))))
DIFF(neg(0), neg(s(s(y_0)))) → COND1(false, 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 COND2(true, neg(0), neg(s(s(x0)))) → DIFF(neg(0), neg(s(x0))) we obtained the following new rules [LPAR04]:

COND2(true, neg(0), neg(s(s(s(y_0))))) → DIFF(neg(0), neg(s(s(y_0))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ 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

                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ MRRProof

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

COND1(false, neg(0), neg(s(s(y_0)))) → COND2(true, neg(0), neg(s(s(y_0))))
DIFF(neg(0), neg(s(s(y_0)))) → COND1(false, neg(0), neg(s(s(y_0))))
COND2(true, neg(0), neg(s(s(s(y_0))))) → DIFF(neg(0), neg(s(s(y_0))))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND1(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(COND2(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(DIFF(x₁, x₂)) = 2 + x₁ + x₂
POL(false) = 0
POL(neg(x₁)) = x₁
POL(s(x₁)) = 3 + x₁
POL(true) = 0

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Rewriting

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ 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

                                                                                                                                              ↳ ForwardInstantiation

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ MRRProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ PisEmptyProof

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