Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar18/ZantemaSLASHz01.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:

a₂(f, a₂(f, a₂(g, a₂(g, x)))) -> a₂(g, a₂(g, a₂(g, a₂(f, a₂(f, a₂(f, x))))))

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

a₂(f, a₂(f, a₂(g, a₂(g, x)))) -> a₂(g, a₂(g, a₂(g, a₂(f, a₂(f, a₂(f, 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:

a₂(f, a₂(f, a₂(g, a₂(g, x)))) -> a₂(g, a₂(g, a₂(g, a₂(f, a₂(f, a₂(f, x))))))

The set Q consists of the following terms:

a₂(f, a₂(f, a₂(g, a₂(g, x0))))

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

A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(g, a₂(g, a₂(f, a₂(f, a₂(f, x)))))
A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(f, a₂(f, x))
A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(g, a₂(f, a₂(f, a₂(f, x))))
A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(f, a₂(f, a₂(f, x)))
A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(f, x)
A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(g, a₂(g, a₂(g, a₂(f, a₂(f, a₂(f, x))))))

The TRS R consists of the following rules:

a₂(f, a₂(f, a₂(g, a₂(g, x)))) -> a₂(g, a₂(g, a₂(g, a₂(f, a₂(f, a₂(f, x))))))

The set Q consists of the following terms:

a₂(f, a₂(f, a₂(g, a₂(g, x0))))

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:

A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(g, a₂(g, a₂(f, a₂(f, a₂(f, x)))))
A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(f, a₂(f, x))
A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(g, a₂(f, a₂(f, a₂(f, x))))
A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(f, a₂(f, a₂(f, x)))
A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(f, x)
A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(g, a₂(g, a₂(g, a₂(f, a₂(f, a₂(f, x))))))

The TRS R consists of the following rules:

a₂(f, a₂(f, a₂(g, a₂(g, x)))) -> a₂(g, a₂(g, a₂(g, a₂(f, a₂(f, a₂(f, x))))))

The set Q consists of the following terms:

a₂(f, a₂(f, a₂(g, a₂(g, x0))))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 3 less nodes.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

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

A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(f, a₂(f, x))
A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(f, a₂(f, a₂(f, x)))
A₂(f, a₂(f, a₂(g, a₂(g, x)))) -> A₂(f, x)

The TRS R consists of the following rules:

a₂(f, a₂(f, a₂(g, a₂(g, x)))) -> a₂(g, a₂(g, a₂(g, a₂(f, a₂(f, a₂(f, x))))))

The set Q consists of the following terms:

a₂(f, a₂(f, a₂(g, a₂(g, x0))))

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