Termination w.r.t. Q proof of /tmp/tmpKfALwD/4.18.trs

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

gcd(x, 0) → x
gcd(0, y) → y
gcd(s(x), s(y)) → if(<(x, y), gcd(s(x), -(y, x)), gcd(-(x, y), s(y)))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

gcd(x, 0) → x
gcd(0, y) → y
gcd(s(x), s(y)) → if(<(x, y), gcd(s(x), -(y, x)), gcd(-(x, y), s(y)))

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

GCD(s(x), s(y)) → GCD(-(x, y), s(y))
GCD(s(x), s(y)) → GCD(s(x), -(y, x))

The TRS R consists of the following rules:

gcd(x, 0) → x
gcd(0, y) → y
gcd(s(x), s(y)) → if(<(x, y), gcd(s(x), -(y, x)), gcd(-(x, y), s(y)))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

GCD(s(x), s(y)) → GCD(-(x, y), s(y))
GCD(s(x), s(y)) → GCD(s(x), -(y, x))

The TRS R consists of the following rules:

gcd(x, 0) → x
gcd(0, y) → y
gcd(s(x), s(y)) → if(<(x, y), gcd(s(x), -(y, x)), gcd(-(x, y), s(y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.