Termination proof of /tmp/tmpZ2xXSQ/gcd

Termination of the given ITRSProblem could successfully be proven:

↳ ITRS

  ↳ ITRStoQTRSProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

Cond_gcd1(TRUE, x, y) → gcd(-@z(y, x), x)
Cond_gcd(TRUE, x, y) → gcd(-@z(x, y), y)
gcd(0@z, y) → y
gcd(x, 0@z) → x
gcd(x, y) → Cond_gcd(&&(>=@z(x, y), >@z(y, 0@z)), x, y)
gcd(x, y) → Cond_gcd1(&&(>@z(y, x), >@z(x, 0@z)), x, y)

The set Q consists of the following terms:

Cond_gcd1(TRUE, x0, x1)
Cond_gcd(TRUE, x0, x1)
gcd(x0, x1)

Represented integers and predefined function symbols by Terms

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

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

Cond_gcd1(true, x, y) → gcd(minus_int(y, x), x)
Cond_gcd(true, x, y) → gcd(minus_int(x, y), y)
gcd(pos(0), y) → y
gcd(x, pos(0)) → x
gcd(x, y) → Cond_gcd(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
gcd(x, y) → Cond_gcd1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
COND_GCD1(true, x, y) → MINUS_INT(y, x)
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
COND_GCD(true, x, y) → MINUS_INT(x, y)
GCD(x, y) → COND_GCD(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
GCD(x, y) → AND(greatereq_int(x, y), greater_int(y, pos(0)))
GCD(x, y) → GREATEREQ_INT(x, y)
GCD(x, y) → GREATER_INT(y, pos(0))
GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
GCD(x, y) → AND(greater_int(y, x), greater_int(x, pos(0)))
GCD(x, y) → GREATER_INT(y, x)
GCD(x, y) → GREATER_INT(x, pos(0))
MINUS_INT(pos(x), pos(y)) → MINUS_NAT(x, y)
MINUS_INT(neg(x), neg(y)) → MINUS_NAT(y, x)
MINUS_INT(neg(x), pos(y)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

The TRS R consists of the following rules:

Cond_gcd1(true, x, y) → gcd(minus_int(y, x), x)
Cond_gcd(true, x, y) → gcd(minus_int(x, y), y)
gcd(pos(0), y) → y
gcd(x, pos(0)) → x
gcd(x, y) → Cond_gcd(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
gcd(x, y) → Cond_gcd1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
COND_GCD1(true, x, y) → MINUS_INT(y, x)
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
COND_GCD(true, x, y) → MINUS_INT(x, y)
GCD(x, y) → COND_GCD(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
GCD(x, y) → AND(greatereq_int(x, y), greater_int(y, pos(0)))
GCD(x, y) → GREATEREQ_INT(x, y)
GCD(x, y) → GREATER_INT(y, pos(0))
GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
GCD(x, y) → AND(greater_int(y, x), greater_int(x, pos(0)))
GCD(x, y) → GREATER_INT(y, x)
GCD(x, y) → GREATER_INT(x, pos(0))
MINUS_INT(pos(x), pos(y)) → MINUS_NAT(x, y)
MINUS_INT(neg(x), neg(y)) → MINUS_NAT(y, x)
MINUS_INT(neg(x), pos(y)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

The TRS R consists of the following rules:

Cond_gcd1(true, x, y) → gcd(minus_int(y, x), x)
Cond_gcd(true, x, y) → gcd(minus_int(x, y), y)
gcd(pos(0), y) → y
gcd(x, pos(0)) → x
gcd(x, y) → Cond_gcd(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
gcd(x, y) → Cond_gcd1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 12 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:

Cond_gcd1(true, x, y) → gcd(minus_int(y, x), x)
Cond_gcd(true, x, y) → gcd(minus_int(x, y), y)
gcd(pos(0), y) → y
gcd(x, pos(0)) → x
gcd(x, y) → Cond_gcd(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
gcd(x, y) → Cond_gcd1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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:

Cond_gcd1(true, x, y) → gcd(minus_int(y, x), x)
Cond_gcd(true, x, y) → gcd(minus_int(x, y), y)
gcd(pos(0), y) → y
gcd(x, pos(0)) → x
gcd(x, y) → Cond_gcd(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
gcd(x, y) → Cond_gcd1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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:

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

The TRS R consists of the following rules:

Cond_gcd1(true, x, y) → gcd(minus_int(y, x), x)
Cond_gcd(true, x, y) → gcd(minus_int(x, y), y)
gcd(pos(0), y) → y
gcd(x, pos(0)) → x
gcd(x, y) → Cond_gcd(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
gcd(x, y) → Cond_gcd1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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:

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

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

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ 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:

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

The TRS R consists of the following rules:

Cond_gcd1(true, x, y) → gcd(minus_int(y, x), x)
Cond_gcd(true, x, y) → gcd(minus_int(x, y), y)
gcd(pos(0), y) → y
gcd(x, pos(0)) → x
gcd(x, y) → Cond_gcd(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
gcd(x, y) → Cond_gcd1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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:

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

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

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ 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:

Cond_gcd1(true, x, y) → gcd(minus_int(y, x), x)
Cond_gcd(true, x, y) → gcd(minus_int(x, y), y)
gcd(pos(0), y) → y
gcd(x, pos(0)) → x
gcd(x, y) → Cond_gcd(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
gcd(x, y) → Cond_gcd1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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:

Cond_gcd1(true, x, y) → gcd(minus_int(y, x), x)
Cond_gcd(true, x, y) → gcd(minus_int(x, y), y)
gcd(pos(0), y) → y
gcd(x, pos(0)) → x
gcd(x, y) → Cond_gcd(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
gcd(x, y) → Cond_gcd1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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:

GCD(x, y) → COND_GCD(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)

The TRS R consists of the following rules:

Cond_gcd1(true, x, y) → gcd(minus_int(y, x), x)
Cond_gcd(true, x, y) → gcd(minus_int(x, y), y)
gcd(pos(0), y) → y
gcd(x, pos(0)) → x
gcd(x, y) → Cond_gcd(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
gcd(x, y) → Cond_gcd1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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:

GCD(x, y) → COND_GCD(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

Cond_gcd1(true, x0, x1)
Cond_gcd(true, x0, x1)
gcd(x0, x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

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

GCD(x, y) → COND_GCD(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y)
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(x, y) → COND_GCD(and(greatereq_int(x, y), greater_int(y, pos(0))), x, y) at position [0] we obtained the following new rules [LPAR04]:

GCD(pos(0), pos(s(x0))) → COND_GCD(and(false, greater_int(pos(s(x0)), pos(0))), pos(0), pos(s(x0)))
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(neg(s(x0)), pos(0)) → COND_GCD(and(false, greater_int(pos(0), pos(0))), neg(s(x0)), pos(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(neg(x0), pos(s(x1))) → COND_GCD(and(false, greater_int(pos(s(x1)), pos(0))), neg(x0), pos(s(x1)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), pos(x0), pos(0))
GCD(neg(s(x0)), neg(0)) → COND_GCD(and(false, greater_int(neg(0), pos(0))), neg(s(x0)), neg(0))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), neg(0), pos(0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x0)), neg(s(x1)))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(0), pos(s(x0))) → COND_GCD(and(false, greater_int(pos(s(x0)), pos(0))), pos(0), pos(s(x0)))
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(neg(s(x0)), pos(0)) → COND_GCD(and(false, greater_int(pos(0), pos(0))), neg(s(x0)), pos(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(neg(x0), pos(s(x1))) → COND_GCD(and(false, greater_int(pos(s(x1)), pos(0))), neg(x0), pos(s(x1)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), pos(x0), pos(0))
GCD(neg(s(x0)), neg(0)) → COND_GCD(and(false, greater_int(neg(0), pos(0))), neg(s(x0)), neg(0))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), neg(0), pos(0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x0)), neg(s(x1)))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

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

GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(0), pos(s(x0))) → COND_GCD(and(false, greater_int(pos(s(x0)), pos(0))), pos(0), pos(s(x0)))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(neg(x0), pos(s(x1))) → COND_GCD(and(false, greater_int(pos(s(x1)), pos(0))), neg(x0), pos(s(x1)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), pos(x0), pos(0))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), neg(0), pos(0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x0)), neg(s(x1)))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(pos(0), pos(s(x0))) → COND_GCD(and(false, greater_int(pos(s(x0)), pos(0))), pos(0), pos(s(x0))) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(pos(0), pos(s(x0))) → COND_GCD(and(false, true), pos(0), pos(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

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

GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(neg(x0), pos(s(x1))) → COND_GCD(and(false, greater_int(pos(s(x1)), pos(0))), neg(x0), pos(s(x1)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), pos(x0), pos(0))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), neg(0), pos(0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x0)), neg(s(x1)))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(0), pos(s(x0))) → COND_GCD(and(false, true), pos(0), pos(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

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

COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(neg(x0), pos(s(x1))) → COND_GCD(and(false, greater_int(pos(s(x1)), pos(0))), neg(x0), pos(s(x1)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), pos(x0), pos(0))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), neg(0), pos(0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x0)), neg(s(x1)))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(neg(x0), pos(s(x1))) → COND_GCD(and(false, greater_int(pos(s(x1)), pos(0))), neg(x0), pos(s(x1))) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(neg(x0), pos(s(x1))) → COND_GCD(and(false, true), neg(x0), pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

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

COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), pos(x0), pos(0))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), neg(0), pos(0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x0)), neg(s(x1)))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(neg(x0), pos(s(x1))) → COND_GCD(and(false, true), neg(x0), pos(s(x1)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

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

GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), pos(x0), pos(0))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), neg(0), pos(0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x0)), neg(s(x1)))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(pos(x0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), pos(x0), pos(0)) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(pos(x0), pos(0)) → COND_GCD(and(true, false), pos(x0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

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

GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), neg(0), pos(0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x0)), neg(s(x1)))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(x0), pos(0)) → COND_GCD(and(true, false), pos(x0), pos(0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

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

COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), neg(0), pos(0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x0)), neg(s(x1)))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(neg(0), pos(0)) → COND_GCD(and(true, greater_int(pos(0), pos(0))), neg(0), pos(0)) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(neg(0), pos(0)) → COND_GCD(and(true, false), neg(0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

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

COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x0)), neg(s(x1)))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(neg(0), pos(0)) → COND_GCD(and(true, false), neg(0), pos(0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

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

GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x0)), neg(s(x1)))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x0)), pos(s(x1))) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

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

GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x0)), neg(s(x1)))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x0)), neg(s(x1))) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

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

GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(x, y) → COND_GCD1(and(greater_int(y, x), greater_int(x, pos(0))), x, y) at position [0] we obtained the following new rules [LPAR04]:

GCD(pos(s(x1)), neg(s(x0))) → COND_GCD1(and(false, greater_int(pos(s(x1)), pos(0))), pos(s(x1)), neg(s(x0)))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x1)), pos(s(x0)))
GCD(pos(0), neg(0)) → COND_GCD1(and(false, greater_int(pos(0), pos(0))), pos(0), neg(0))
GCD(neg(s(x0)), neg(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), neg(0))
GCD(neg(s(x1)), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(s(x1)), pos(0))), neg(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x1)), neg(s(x0)))
GCD(neg(0), pos(0)) → COND_GCD1(and(false, greater_int(neg(0), pos(0))), neg(0), pos(0))
GCD(pos(0), pos(0)) → COND_GCD1(and(false, greater_int(pos(0), pos(0))), pos(0), pos(0))
GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(pos(0), pos(0))), pos(0), pos(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(neg(0), neg(0)) → COND_GCD1(and(false, greater_int(neg(0), pos(0))), neg(0), neg(0))
GCD(pos(0), neg(s(x0))) → COND_GCD1(and(false, greater_int(pos(0), pos(0))), pos(0), neg(s(x0)))
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0))
GCD(neg(0), neg(s(x0))) → COND_GCD1(and(false, greater_int(neg(0), pos(0))), neg(0), neg(s(x0)))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

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

COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(pos(s(x1)), neg(s(x0))) → COND_GCD1(and(false, greater_int(pos(s(x1)), pos(0))), pos(s(x1)), neg(s(x0)))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x1)), pos(s(x0)))
GCD(pos(0), neg(0)) → COND_GCD1(and(false, greater_int(pos(0), pos(0))), pos(0), neg(0))
GCD(neg(s(x0)), neg(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), neg(0))
GCD(neg(s(x1)), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(s(x1)), pos(0))), neg(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x1)), neg(s(x0)))
GCD(neg(0), pos(0)) → COND_GCD1(and(false, greater_int(neg(0), pos(0))), neg(0), pos(0))
GCD(pos(0), pos(0)) → COND_GCD1(and(false, greater_int(pos(0), pos(0))), pos(0), pos(0))
GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(pos(0), pos(0))), pos(0), pos(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(neg(0), neg(0)) → COND_GCD1(and(false, greater_int(neg(0), pos(0))), neg(0), neg(0))
GCD(pos(0), neg(s(x0))) → COND_GCD1(and(false, greater_int(pos(0), pos(0))), pos(0), neg(s(x0)))
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0))
GCD(neg(0), neg(s(x0))) → COND_GCD1(and(false, greater_int(neg(0), pos(0))), neg(0), neg(s(x0)))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 6 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

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

GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(pos(s(x1)), neg(s(x0))) → COND_GCD1(and(false, greater_int(pos(s(x1)), pos(0))), pos(s(x1)), neg(s(x0)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x0)), neg(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), neg(0))
GCD(neg(s(x1)), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(s(x1)), pos(0))), neg(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x1)), neg(s(x0)))
GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(pos(0), pos(0))), pos(0), pos(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(pos(s(x1)), neg(s(x0))) → COND_GCD1(and(false, greater_int(pos(s(x1)), pos(0))), pos(s(x1)), neg(s(x0))) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(pos(s(x1)), neg(s(x0))) → COND_GCD1(and(false, true), pos(s(x1)), neg(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

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

GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x0)), neg(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), neg(0))
GCD(neg(s(x1)), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(s(x1)), pos(0))), neg(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x1)), neg(s(x0)))
GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(pos(0), pos(0))), pos(0), pos(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), neg(s(x0))) → COND_GCD1(and(false, true), pos(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x1)), pos(s(x0)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x0)), neg(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), neg(0))
GCD(neg(s(x1)), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(s(x1)), pos(0))), neg(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x1)), neg(s(x0)))
GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(pos(0), pos(0))), pos(0), pos(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(0))), pos(s(x1)), pos(s(x0))) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x0)), neg(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), neg(0))
GCD(neg(s(x1)), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(s(x1)), pos(0))), neg(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x1)), neg(s(x0)))
GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(pos(0), pos(0))), pos(0), pos(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(neg(s(x0)), neg(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), neg(0)) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(neg(s(x0)), neg(0)) → COND_GCD1(and(true, false), neg(s(x0)), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x1)), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(s(x1)), pos(0))), neg(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x1)), neg(s(x0)))
GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(pos(0), pos(0))), pos(0), pos(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x0)), neg(0)) → COND_GCD1(and(true, false), neg(s(x0)), neg(0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

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

GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(s(x1)), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(s(x1)), pos(0))), neg(s(x1)), pos(s(x0)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x1)), neg(s(x0)))
GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(pos(0), pos(0))), pos(0), pos(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(neg(s(x1)), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(s(x1)), pos(0))), neg(s(x1)), pos(s(x0))) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(neg(s(x1)), pos(s(x0))) → COND_GCD1(and(true, false), neg(s(x1)), pos(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

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

GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x1)), neg(s(x0)))
GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(pos(0), pos(0))), pos(0), pos(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), pos(s(x0))) → COND_GCD1(and(true, false), neg(s(x1)), pos(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x1)), neg(s(x0)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(pos(0), pos(0))), pos(0), pos(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(0))), neg(s(x1)), neg(s(x0))) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(pos(0), pos(0))), pos(0), pos(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(pos(0), pos(0))), pos(0), pos(s(x0))) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, false), pos(0), pos(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, false), pos(0), pos(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

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

GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), pos(0)) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, true), pos(s(x0)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

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

GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, true), pos(s(x0)), pos(0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, greater_int(neg(s(x0)), pos(0))), neg(s(x0)), pos(0)) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, false), neg(s(x0)), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, false), neg(s(x0)), pos(0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

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

GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, greater_int(neg(0), pos(0))), neg(0), pos(s(x0))) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, false), neg(0), pos(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

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

GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), pos(s(x0))) → COND_GCD1(and(true, false), neg(0), pos(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, greater_int(pos(s(x0)), pos(0))), pos(s(x0)), neg(0)) at position [0,1] we obtained the following new rules [LPAR04]:

GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, true), pos(s(x0)), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, true), pos(s(x0)), neg(0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

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

GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(y0, pos(0)) → COND_GCD(and(greatereq_int(y0, pos(0)), false), y0, pos(0)) at position [0] we obtained the following new rules [LPAR04]:

GCD(neg(0), pos(0)) → COND_GCD(and(true, false), neg(0), pos(0))
GCD(neg(s(x0)), pos(0)) → COND_GCD(and(false, false), neg(s(x0)), pos(0))
GCD(pos(x0), pos(0)) → COND_GCD(and(true, false), pos(x0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), pos(0)) → COND_GCD(and(true, false), neg(0), pos(0))
GCD(neg(s(x0)), pos(0)) → COND_GCD(and(false, false), neg(s(x0)), pos(0))
GCD(pos(x0), pos(0)) → COND_GCD(and(true, false), pos(x0), pos(0))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

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

GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

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

GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(y0, neg(0)) → COND_GCD(and(greatereq_int(y0, neg(0)), false), y0, neg(0)) at position [0] we obtained the following new rules [LPAR04]:

GCD(neg(s(x0)), neg(0)) → COND_GCD(and(false, false), neg(s(x0)), neg(0))
GCD(neg(0), neg(0)) → COND_GCD(and(true, false), neg(0), neg(0))
GCD(pos(x0), neg(0)) → COND_GCD(and(true, false), pos(x0), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(s(x0)), neg(0)) → COND_GCD(and(false, false), neg(s(x0)), neg(0))
GCD(neg(0), neg(0)) → COND_GCD(and(true, false), neg(0), neg(0))
GCD(pos(x0), neg(0)) → COND_GCD(and(true, false), pos(x0), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

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

GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0)))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(y0, pos(s(x0))) → COND_GCD(and(greatereq_int(y0, pos(s(x0))), true), y0, pos(s(x0))) at position [0] we obtained the following new rules [LPAR04]:

GCD(neg(x0), pos(s(x1))) → COND_GCD(and(false, true), neg(x0), pos(s(x1)))
GCD(pos(0), pos(s(x0))) → COND_GCD(and(false, true), pos(0), pos(s(x0)))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(x0), pos(s(x1))) → COND_GCD(and(false, true), neg(x0), pos(s(x1)))
GCD(pos(0), pos(s(x0))) → COND_GCD(and(false, true), pos(0), pos(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

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

GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

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

GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(y0, neg(s(x0))) → COND_GCD(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0))) at position [0] we obtained the following new rules [LPAR04]:

GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), neg(s(y1))) → COND_GCD(and(true, false), neg(0), neg(s(y1)))
GCD(pos(x0), neg(s(y1))) → COND_GCD(and(true, false), pos(x0), neg(s(y1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), neg(s(y1))) → COND_GCD(and(true, false), neg(0), neg(s(y1)))
GCD(pos(x0), neg(s(y1))) → COND_GCD(and(true, false), pos(x0), neg(s(y1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

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

GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

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

GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(pos(x0), neg(x1)) → COND_GCD(and(true, greater_int(neg(x1), pos(0))), pos(x0), neg(x1)) at position [0] we obtained the following new rules [LPAR04]:

GCD(pos(y0), neg(s(x0))) → COND_GCD(and(true, false), pos(y0), neg(s(x0)))
GCD(pos(y0), neg(0)) → COND_GCD(and(true, false), pos(y0), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(pos(y0), neg(s(x0))) → COND_GCD(and(true, false), pos(y0), neg(s(x0)))
GCD(pos(y0), neg(0)) → COND_GCD(and(true, false), pos(y0), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

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

GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(neg(0), neg(x0)) → COND_GCD(and(true, greater_int(neg(x0), pos(0))), neg(0), neg(x0)) at position [0] we obtained the following new rules [LPAR04]:

GCD(neg(0), neg(s(x0))) → COND_GCD(and(true, false), neg(0), neg(s(x0)))
GCD(neg(0), neg(0)) → COND_GCD(and(true, false), neg(0), neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

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

COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), neg(s(x0))) → COND_GCD(and(true, false), neg(0), neg(s(x0)))
GCD(neg(0), neg(0)) → COND_GCD(and(true, false), neg(0), neg(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

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

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
COND_GCD(true, x, y) → GCD(minus_int(x, y), y)
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

COND_GCD(true, pos(x0), neg(x1)) → GCD(pos(plus_nat(x0, x1)), neg(x1))
COND_GCD(true, neg(x0), pos(x1)) → GCD(neg(plus_nat(x0, x1)), pos(x1))
COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

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

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
COND_GCD(true, pos(x0), neg(x1)) → GCD(pos(plus_nat(x0, x1)), neg(x1))
COND_GCD(true, neg(x0), pos(x1)) → GCD(neg(plus_nat(x0, x1)), pos(x1))
COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 2 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

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

COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(pos(s(x0)), y1) → COND_GCD1(and(greater_int(y1, pos(s(x0))), true), pos(s(x0)), y1) at position [0] we obtained the following new rules [LPAR04]:

GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, true), pos(s(x0)), neg(0))
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, true), pos(s(x0)), pos(0))
GCD(pos(s(x1)), neg(s(x0))) → COND_GCD1(and(false, true), pos(s(x1)), neg(s(x0)))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

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

COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(pos(s(x0)), neg(0)) → COND_GCD1(and(false, true), pos(s(x0)), neg(0))
GCD(pos(s(x0)), pos(0)) → COND_GCD1(and(false, true), pos(s(x0)), pos(0))
GCD(pos(s(x1)), neg(s(x0))) → COND_GCD1(and(false, true), pos(s(x1)), neg(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

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

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

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

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(neg(s(x0)), y1) → COND_GCD1(and(greater_int(y1, neg(s(x0))), false), neg(s(x0)), y1) at position [0] we obtained the following new rules [LPAR04]:

GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, false), neg(s(x0)), pos(0))
GCD(neg(s(x0)), neg(0)) → COND_GCD1(and(true, false), neg(s(x0)), neg(0))
GCD(neg(s(x1)), pos(s(x0))) → COND_GCD1(and(true, false), neg(s(x1)), pos(s(x0)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

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

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x0)), pos(0)) → COND_GCD1(and(true, false), neg(s(x0)), pos(0))
GCD(neg(s(x0)), neg(0)) → COND_GCD1(and(true, false), neg(s(x0)), neg(0))
GCD(neg(s(x1)), pos(s(x0))) → COND_GCD1(and(true, false), neg(s(x1)), pos(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

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

COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

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

COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1)
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 GCD(pos(0), y1) → COND_GCD1(and(greater_int(y1, pos(0)), false), pos(0), y1) at position [0] we obtained the following new rules [LPAR04]:

GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, false), pos(0), pos(s(x0)))
GCD(pos(0), neg(0)) → COND_GCD1(and(false, false), pos(0), neg(0))
GCD(pos(0), neg(s(x0))) → COND_GCD1(and(false, false), pos(0), neg(s(x0)))
GCD(pos(0), pos(0)) → COND_GCD1(and(false, false), pos(0), pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

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

COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(pos(0), pos(s(x0))) → COND_GCD1(and(true, false), pos(0), pos(s(x0)))
GCD(pos(0), neg(0)) → COND_GCD1(and(false, false), pos(0), neg(0))
GCD(pos(0), neg(s(x0))) → COND_GCD1(and(false, false), pos(0), neg(s(x0)))
GCD(pos(0), pos(0)) → COND_GCD1(and(false, false), pos(0), pos(0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

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

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

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

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
COND_GCD1(true, x, y) → GCD(minus_int(y, x), x)
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_GCD1(true, x, y) → GCD(minus_int(y, x), x) at position [0] we obtained the following new rules [LPAR04]:

COND_GCD1(true, neg(x1), pos(x0)) → GCD(pos(plus_nat(x0, x1)), neg(x1))
COND_GCD1(true, pos(x1), neg(x0)) → GCD(neg(plus_nat(x0, x1)), pos(x1))
COND_GCD1(true, neg(x1), neg(x0)) → GCD(minus_nat(x1, x0), neg(x1))
COND_GCD1(true, pos(x1), pos(x0)) → GCD(minus_nat(x0, x1), pos(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

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

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
GCD(neg(0), y1) → COND_GCD1(and(greater_int(y1, neg(0)), false), neg(0), y1)
COND_GCD1(true, neg(x1), pos(x0)) → GCD(pos(plus_nat(x0, x1)), neg(x1))
COND_GCD1(true, pos(x1), neg(x0)) → GCD(neg(plus_nat(x0, x1)), pos(x1))
COND_GCD1(true, neg(x1), neg(x0)) → GCD(minus_nat(x1, x0), neg(x1))
COND_GCD1(true, pos(x1), pos(x0)) → GCD(minus_nat(x0, x1), pos(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                        ↳ QDP

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

COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
COND_GCD1(true, neg(x1), neg(x0)) → GCD(minus_nat(x1, x0), neg(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                        ↳ QDP

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

COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
COND_GCD1(true, neg(x1), neg(x0)) → GCD(minus_nat(x1, x0), neg(x1))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                        ↳ QDP

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

COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1))
GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
COND_GCD1(true, neg(x1), neg(x0)) → GCD(minus_nat(x1, x0), neg(x1))

The TRS R consists of the following rules:

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

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_GCD(true, neg(x0), neg(x1)) → GCD(minus_nat(x1, x0), neg(x1)) we obtained the following new rules [LPAR04]:

COND_GCD(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(s(z1), s(z0)), neg(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                                                        ↳ QDP

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

GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
COND_GCD1(true, neg(x1), neg(x0)) → GCD(minus_nat(x1, x0), neg(x1))
COND_GCD(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(s(z1), s(z0)), neg(s(z1)))

The TRS R consists of the following rules:

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

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

COND_GCD(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(z1, z0), neg(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                                                                        ↳ QDP

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

GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
COND_GCD1(true, neg(x1), neg(x0)) → GCD(minus_nat(x1, x0), neg(x1))
COND_GCD(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(z1, z0), neg(s(z1)))

The TRS R consists of the following rules:

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

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_GCD1(true, neg(x1), neg(x0)) → GCD(minus_nat(x1, x0), neg(x1)) we obtained the following new rules [LPAR04]:

COND_GCD1(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(s(z0), s(z1)), neg(s(z0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                                                        ↳ QDP

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

GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
COND_GCD(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(z1, z0), neg(s(z1)))
COND_GCD1(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(s(z0), s(z1)), neg(s(z0)))

The TRS R consists of the following rules:

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

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_GCD1(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(s(z0), s(z1)), neg(s(z0))) at position [0] we obtained the following new rules [LPAR04]:

COND_GCD1(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(z0, z1), neg(s(z0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                                                        ↳ QDP

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

GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
COND_GCD(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(z1, z0), neg(s(z1)))
COND_GCD1(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(z0, z1), neg(s(z0)))

The TRS R consists of the following rules:

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

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

The following pairs can be oriented strictly and are deleted.

COND_GCD(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(z1, z0), neg(s(z1)))

The remaining pairs can at least be oriented weakly.

GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
COND_GCD1(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(z0, z1), neg(s(z0)))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND_GCD(x₁, x₂, x₃)) = x₁
POL(COND_GCD1(x₁, x₂, x₃)) = 0
POL(GCD(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 0
POL(false) = 0
POL(greater_int(x₁, x₂)) = 0
POL(greatereq_int(x₁, x₂)) = x₂
POL(minus_nat(x₁, x₂)) = 0
POL(neg(x₁)) = 1
POL(pos(x₁)) = 0
POL(s(x₁)) = 0
POL(true) = 1

The following usable rules [FROCOS05] were oriented:

and(false, false) → false
and(true, false) → false

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                        ↳ QDP

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

GCD(neg(s(x0)), neg(s(x1))) → COND_GCD(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
COND_GCD1(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(z0, z1), neg(s(z0)))

The TRS R consists of the following rules:

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

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                        ↳ QDP

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

GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
COND_GCD1(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(z0, z1), neg(s(z0)))

The TRS R consists of the following rules:

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

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                        ↳ QDP

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

GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
COND_GCD1(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(z0, z1), neg(s(z0)))

The TRS R consists of the following rules:

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(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))
and(false, false) → false
and(true, false) → false

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))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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].

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                                                        ↳ QDP

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

GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))
COND_GCD1(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(z0, z1), neg(s(z0)))

The TRS R consists of the following rules:

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(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))
and(false, false) → false
and(true, false) → false

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))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

The following pairs can be oriented strictly and are deleted.

COND_GCD1(true, neg(s(z0)), neg(s(z1))) → GCD(minus_nat(z0, z1), neg(s(z0)))

The remaining pairs can at least be oriented weakly.

GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), neg(s(x0)))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1
POL(COND_GCD1(x₁, x₂, x₃)) = x₁ + x₃
POL(GCD(x₁, x₂)) = 1
POL(and(x₁, x₂)) = 0
POL(false) = 0
POL(greater_int(x₁, x₂)) = 1 + x₂
POL(minus_nat(x₁, x₂)) = x₁
POL(neg(x₁)) = 1
POL(pos(x₁)) = 0
POL(s(x₁)) = x₁
POL(true) = 1

The following usable rules [FROCOS05] were oriented:

and(true, false) → false
and(false, false) → false

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                        ↳ QDP

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

GCD(neg(s(x1)), neg(s(x0))) → COND_GCD1(and(greater_int(neg(x0), neg(x1)), false), neg(s(x1)), 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))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false

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))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

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

COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
COND_GCD1(true, pos(x1), pos(x0)) → GCD(minus_nat(x0, x1), pos(x1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

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

COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
COND_GCD1(true, pos(x1), pos(x0)) → GCD(minus_nat(x0, x1), pos(x1))

The TRS R consists of the following rules:

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

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

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

COND_GCD(true, pos(x0), pos(x1)) → GCD(minus_nat(x0, x1), pos(x1))
GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
COND_GCD1(true, pos(x1), pos(x0)) → GCD(minus_nat(x0, x1), pos(x1))

The TRS R consists of the following rules:

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

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

COND_GCD(true, pos(s(z0)), pos(s(z1))) → GCD(minus_nat(s(z0), s(z1)), pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

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

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
COND_GCD1(true, pos(x1), pos(x0)) → GCD(minus_nat(x0, x1), pos(x1))
COND_GCD(true, pos(s(z0)), pos(s(z1))) → GCD(minus_nat(s(z0), s(z1)), pos(s(z1)))

The TRS R consists of the following rules:

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

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

COND_GCD(true, pos(s(z0)), pos(s(z1))) → GCD(minus_nat(z0, z1), pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ Instantiation

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

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
COND_GCD1(true, pos(x1), pos(x0)) → GCD(minus_nat(x0, x1), pos(x1))
COND_GCD(true, pos(s(z0)), pos(s(z1))) → GCD(minus_nat(z0, z1), pos(s(z1)))

The TRS R consists of the following rules:

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

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_GCD1(true, pos(x1), pos(x0)) → GCD(minus_nat(x0, x1), pos(x1)) we obtained the following new rules [LPAR04]:

COND_GCD1(true, pos(s(z0)), pos(s(z1))) → GCD(minus_nat(s(z1), s(z0)), pos(s(z0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Rewriting

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

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
COND_GCD(true, pos(s(z0)), pos(s(z1))) → GCD(minus_nat(z0, z1), pos(s(z1)))
COND_GCD1(true, pos(s(z0)), pos(s(z1))) → GCD(minus_nat(s(z1), s(z0)), pos(s(z0)))

The TRS R consists of the following rules:

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

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_GCD1(true, pos(s(z0)), pos(s(z1))) → GCD(minus_nat(s(z1), s(z0)), pos(s(z0))) at position [0] we obtained the following new rules [LPAR04]:

COND_GCD1(true, pos(s(z0)), pos(s(z1))) → GCD(minus_nat(z1, z0), pos(s(z0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ QDPOrderProof

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

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))
COND_GCD(true, pos(s(z0)), pos(s(z1))) → GCD(minus_nat(z0, z1), pos(s(z1)))
COND_GCD1(true, pos(s(z0)), pos(s(z1))) → GCD(minus_nat(z1, z0), pos(s(z0)))

The TRS R consists of the following rules:

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

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

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

The following pairs can be oriented strictly and are deleted.

COND_GCD(true, pos(s(z0)), pos(s(z1))) → GCD(minus_nat(z0, z1), pos(s(z1)))
COND_GCD1(true, pos(s(z0)), pos(s(z1))) → GCD(minus_nat(z1, z0), pos(s(z0)))

The remaining pairs can at least be oriented weakly.

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND_GCD(x₁, x₂, x₃)) = x₂ + x₃
POL(COND_GCD1(x₁, x₂, x₃)) = x₂ + x₃
POL(GCD(x₁, x₂)) = x₁ + x₂
POL(and(x₁, x₂)) = 0
POL(false) = 0
POL(greater_int(x₁, x₂)) = 0
POL(greatereq_int(x₁, x₂)) = 0
POL(minus_nat(x₁, x₂)) = x₁
POL(neg(x₁)) = 0
POL(pos(x₁)) = x₁
POL(s(x₁)) = 1 + x₁
POL(true) = 0

The following usable rules [FROCOS05] were oriented:

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)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Rewriting

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Rewriting

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Rewriting

                                                                                                              ↳ QDP

                                                                                                                ↳ Rewriting

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Rewriting

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Rewriting

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Rewriting

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Narrowing

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                            ↳ Narrowing

                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                    ↳ UsableRulesProof

                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                        ↳ Narrowing

                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                ↳ UsableRulesProof

                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                    ↳ Narrowing

                                                                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                                                                            ↳ UsableRulesProof

                                                                                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                                                                                ↳ Narrowing

                                                                                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                      ↳ AND

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ UsableRulesProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ QReductionProof

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ Instantiation

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Rewriting

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ Instantiation

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Rewriting

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                      ↳ DependencyGraphProof

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

GCD(pos(s(x0)), pos(s(x1))) → COND_GCD(and(greatereq_int(pos(x0), pos(x1)), true), pos(s(x0)), pos(s(x1)))
GCD(pos(s(x1)), pos(s(x0))) → COND_GCD1(and(greater_int(pos(x0), pos(x1)), true), pos(s(x1)), pos(s(x0)))

The TRS R consists of the following rules:

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

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
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 2 less nodes.