Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar3/ZantemaSLASHjw09.trs

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

f₂(a, f₂(a, x)) -> f₂(x, f₂(a, f₂(f₂(a, a), a)))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₂(a, f₂(a, x)) -> f₂(x, f₂(a, f₂(f₂(a, a), a)))

Q is empty.

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

F₂(a, f₂(a, x)) -> F₂(a, a)
F₂(a, f₂(a, x)) -> F₂(x, f₂(a, f₂(f₂(a, a), a)))
F₂(a, f₂(a, x)) -> F₂(f₂(a, a), a)
F₂(a, f₂(a, x)) -> F₂(a, f₂(f₂(a, a), a))

The TRS R consists of the following rules:

f₂(a, f₂(a, x)) -> f₂(x, f₂(a, f₂(f₂(a, a), a)))

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:

F₂(a, f₂(a, x)) -> F₂(a, a)
F₂(a, f₂(a, x)) -> F₂(x, f₂(a, f₂(f₂(a, a), a)))
F₂(a, f₂(a, x)) -> F₂(f₂(a, a), a)
F₂(a, f₂(a, x)) -> F₂(a, f₂(f₂(a, a), a))

The TRS R consists of the following rules:

f₂(a, f₂(a, x)) -> f₂(x, f₂(a, f₂(f₂(a, a), a)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 1 SCC with 3 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

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

F₂(a, f₂(a, x)) -> F₂(x, f₂(a, f₂(f₂(a, a), a)))

The TRS R consists of the following rules:

f₂(a, f₂(a, x)) -> f₂(x, f₂(a, f₂(f₂(a, a), a)))

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