Termination w.r.t. Q proof of TRS/various09.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:

f₅(x, y, w, w, a) -> g1₄(x, x, y, w)
f₅(x, y, w, a, a) -> g1₄(y, x, x, w)
f₅(x, y, a, a, w) -> g2₄(x, y, y, w)
f₅(x, y, a, w, w) -> g2₄(y, y, x, w)
g1₄(x, x, y, a) -> h₂(x, y)
g1₄(y, x, x, a) -> h₂(x, y)
g2₄(x, y, y, a) -> h₂(x, y)
g2₄(y, y, x, a) -> h₂(x, y)
h₂(x, x) -> x

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₅(x, y, w, w, a) -> g1₄(x, x, y, w)
f₅(x, y, w, a, a) -> g1₄(y, x, x, w)
f₅(x, y, a, a, w) -> g2₄(x, y, y, w)
f₅(x, y, a, w, w) -> g2₄(y, y, x, w)
g1₄(x, x, y, a) -> h₂(x, y)
g1₄(y, x, x, a) -> h₂(x, y)
g2₄(x, y, y, a) -> h₂(x, y)
g2₄(y, y, x, a) -> h₂(x, y)
h₂(x, x) -> x

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

F₅(x, y, a, a, w) -> G2₄(x, y, y, w)
F₅(x, y, w, w, a) -> G1₄(x, x, y, w)
G1₄(y, x, x, a) -> H₂(x, y)
G1₄(x, x, y, a) -> H₂(x, y)
G2₄(y, y, x, a) -> H₂(x, y)
F₅(x, y, w, a, a) -> G1₄(y, x, x, w)
F₅(x, y, a, w, w) -> G2₄(y, y, x, w)
G2₄(x, y, y, a) -> H₂(x, y)

The TRS R consists of the following rules:

f₅(x, y, w, w, a) -> g1₄(x, x, y, w)
f₅(x, y, w, a, a) -> g1₄(y, x, x, w)
f₅(x, y, a, a, w) -> g2₄(x, y, y, w)
f₅(x, y, a, w, w) -> g2₄(y, y, x, w)
g1₄(x, x, y, a) -> h₂(x, y)
g1₄(y, x, x, a) -> h₂(x, y)
g2₄(x, y, y, a) -> h₂(x, y)
g2₄(y, y, x, a) -> h₂(x, y)
h₂(x, x) -> x

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:

F₅(x, y, a, a, w) -> G2₄(x, y, y, w)
F₅(x, y, w, w, a) -> G1₄(x, x, y, w)
G1₄(y, x, x, a) -> H₂(x, y)
G1₄(x, x, y, a) -> H₂(x, y)
G2₄(y, y, x, a) -> H₂(x, y)
F₅(x, y, w, a, a) -> G1₄(y, x, x, w)
F₅(x, y, a, w, w) -> G2₄(y, y, x, w)
G2₄(x, y, y, a) -> H₂(x, y)

The TRS R consists of the following rules:

f₅(x, y, w, w, a) -> g1₄(x, x, y, w)
f₅(x, y, w, a, a) -> g1₄(y, x, x, w)
f₅(x, y, a, a, w) -> g2₄(x, y, y, w)
f₅(x, y, a, w, w) -> g2₄(y, y, x, w)
g1₄(x, x, y, a) -> h₂(x, y)
g1₄(y, x, x, a) -> h₂(x, y)
g2₄(x, y, y, a) -> h₂(x, y)
g2₄(y, y, x, a) -> h₂(x, y)
h₂(x, x) -> x

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 8 less nodes.