Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar1/HofWaldSLASH5.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(x, f₂(a, a)) -> f₂(f₂(x, a), x)

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

f₂(x, f₂(a, a)) -> f₂(f₂(x, a), x)

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

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

f₂(x, f₂(a, a)) -> f₂(f₂(x, a), x)

The set Q consists of the following terms:

f₂(x0, f₂(a, a))

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

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

The TRS R consists of the following rules:

f₂(x, f₂(a, a)) -> f₂(f₂(x, a), x)

The set Q consists of the following terms:

f₂(x0, f₂(a, a))

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

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

The TRS R consists of the following rules:

f₂(x, f₂(a, a)) -> f₂(f₂(x, a), x)

The set Q consists of the following terms:

f₂(x0, f₂(a, a))

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

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

F₂(x, f₂(a, a)) -> F₂(f₂(x, a), x)

The TRS R consists of the following rules:

f₂(x, f₂(a, a)) -> f₂(f₂(x, a), x)

The set Q consists of the following terms:

f₂(x0, f₂(a, a))

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