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))))


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 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.