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:

a(a(x1)) → b(x1)
a(a(a(x1))) → a(b(a(x1)))
a(b(a(x1))) → b(b(b(x1)))
a(a(a(a(x1)))) → a(a(b(a(a(x1)))))
a(a(b(a(x1)))) → a(b(b(a(b(x1)))))
a(b(a(a(x1)))) → b(a(b(b(a(x1)))))
a(b(b(a(x1)))) → b(b(b(b(b(x1)))))
a(a(a(a(a(x1))))) → a(a(a(b(a(a(a(x1)))))))
a(a(a(b(a(x1))))) → a(a(b(b(a(a(b(x1)))))))
a(a(b(a(a(x1))))) → a(b(a(b(a(b(a(x1)))))))
a(a(b(b(a(x1))))) → a(b(b(b(a(b(b(x1)))))))
a(b(a(a(a(x1))))) → b(a(a(b(b(a(a(x1)))))))
a(b(a(b(a(x1))))) → b(a(b(b(b(a(b(x1)))))))
a(b(b(a(a(x1))))) → b(b(a(b(b(b(a(x1)))))))
a(b(b(b(a(x1))))) → b(b(b(b(b(b(b(x1)))))))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

a(a(x1)) → b(x1)
a(a(a(x1))) → a(b(a(x1)))
a(b(a(x1))) → b(b(b(x1)))
a(a(a(a(x1)))) → a(a(b(a(a(x1)))))
a(a(b(a(x1)))) → a(b(b(a(b(x1)))))
a(b(a(a(x1)))) → b(a(b(b(a(x1)))))
a(b(b(a(x1)))) → b(b(b(b(b(x1)))))
a(a(a(a(a(x1))))) → a(a(a(b(a(a(a(x1)))))))
a(a(a(b(a(x1))))) → a(a(b(b(a(a(b(x1)))))))
a(a(b(a(a(x1))))) → a(b(a(b(a(b(a(x1)))))))
a(a(b(b(a(x1))))) → a(b(b(b(a(b(b(x1)))))))
a(b(a(a(a(x1))))) → b(a(a(b(b(a(a(x1)))))))
a(b(a(b(a(x1))))) → b(a(b(b(b(a(b(x1)))))))
a(b(b(a(a(x1))))) → b(b(a(b(b(b(a(x1)))))))
a(b(b(b(a(x1))))) → b(b(b(b(b(b(b(x1)))))))

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

A(b(b(a(a(x1))))) → A(b(b(b(a(x1)))))
A(a(b(b(a(x1))))) → A(b(b(b(a(b(b(x1)))))))
A(a(a(a(x1)))) → A(a(b(a(a(x1)))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(x1))))) → A(a(b(b(a(a(x1))))))
A(a(b(a(x1)))) → A(b(x1))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(a(a(a(a(x1))))) → A(b(a(a(a(x1)))))
A(a(a(b(a(x1))))) → A(b(x1))
A(b(a(b(a(x1))))) → A(b(b(b(a(b(x1))))))
A(b(a(a(a(x1))))) → A(b(b(a(a(x1)))))
A(a(b(b(a(x1))))) → A(b(b(x1)))
A(a(a(a(a(x1))))) → A(a(b(a(a(a(x1))))))
A(a(a(b(a(x1))))) → A(a(b(x1)))
A(b(a(b(a(x1))))) → A(b(x1))
A(a(a(x1))) → A(b(a(x1)))
A(a(a(b(a(x1))))) → A(b(b(a(a(b(x1))))))
A(a(a(a(x1)))) → A(b(a(a(x1))))
A(a(a(a(a(x1))))) → A(a(a(b(a(a(a(x1)))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(b(a(a(x1)))) → A(b(b(a(x1))))
A(a(b(a(x1)))) → A(b(b(a(b(x1)))))
A(a(a(b(a(x1))))) → A(a(b(b(a(a(b(x1)))))))

The TRS R consists of the following rules:

a(a(x1)) → b(x1)
a(a(a(x1))) → a(b(a(x1)))
a(b(a(x1))) → b(b(b(x1)))
a(a(a(a(x1)))) → a(a(b(a(a(x1)))))
a(a(b(a(x1)))) → a(b(b(a(b(x1)))))
a(b(a(a(x1)))) → b(a(b(b(a(x1)))))
a(b(b(a(x1)))) → b(b(b(b(b(x1)))))
a(a(a(a(a(x1))))) → a(a(a(b(a(a(a(x1)))))))
a(a(a(b(a(x1))))) → a(a(b(b(a(a(b(x1)))))))
a(a(b(a(a(x1))))) → a(b(a(b(a(b(a(x1)))))))
a(a(b(b(a(x1))))) → a(b(b(b(a(b(b(x1)))))))
a(b(a(a(a(x1))))) → b(a(a(b(b(a(a(x1)))))))
a(b(a(b(a(x1))))) → b(a(b(b(b(a(b(x1)))))))
a(b(b(a(a(x1))))) → b(b(a(b(b(b(a(x1)))))))
a(b(b(b(a(x1))))) → b(b(b(b(b(b(b(x1)))))))

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:

A(b(b(a(a(x1))))) → A(b(b(b(a(x1)))))
A(a(b(b(a(x1))))) → A(b(b(b(a(b(b(x1)))))))
A(a(a(a(x1)))) → A(a(b(a(a(x1)))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(x1))))) → A(a(b(b(a(a(x1))))))
A(a(b(a(x1)))) → A(b(x1))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(a(a(a(a(x1))))) → A(b(a(a(a(x1)))))
A(a(a(b(a(x1))))) → A(b(x1))
A(b(a(b(a(x1))))) → A(b(b(b(a(b(x1))))))
A(b(a(a(a(x1))))) → A(b(b(a(a(x1)))))
A(a(b(b(a(x1))))) → A(b(b(x1)))
A(a(a(a(a(x1))))) → A(a(b(a(a(a(x1))))))
A(a(a(b(a(x1))))) → A(a(b(x1)))
A(b(a(b(a(x1))))) → A(b(x1))
A(a(a(x1))) → A(b(a(x1)))
A(a(a(b(a(x1))))) → A(b(b(a(a(b(x1))))))
A(a(a(a(x1)))) → A(b(a(a(x1))))
A(a(a(a(a(x1))))) → A(a(a(b(a(a(a(x1)))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(b(a(a(x1)))) → A(b(b(a(x1))))
A(a(b(a(x1)))) → A(b(b(a(b(x1)))))
A(a(a(b(a(x1))))) → A(a(b(b(a(a(b(x1)))))))

The TRS R consists of the following rules:

a(a(x1)) → b(x1)
a(a(a(x1))) → a(b(a(x1)))
a(b(a(x1))) → b(b(b(x1)))
a(a(a(a(x1)))) → a(a(b(a(a(x1)))))
a(a(b(a(x1)))) → a(b(b(a(b(x1)))))
a(b(a(a(x1)))) → b(a(b(b(a(x1)))))
a(b(b(a(x1)))) → b(b(b(b(b(x1)))))
a(a(a(a(a(x1))))) → a(a(a(b(a(a(a(x1)))))))
a(a(a(b(a(x1))))) → a(a(b(b(a(a(b(x1)))))))
a(a(b(a(a(x1))))) → a(b(a(b(a(b(a(x1)))))))
a(a(b(b(a(x1))))) → a(b(b(b(a(b(b(x1)))))))
a(b(a(a(a(x1))))) → b(a(a(b(b(a(a(x1)))))))
a(b(a(b(a(x1))))) → b(a(b(b(b(a(b(x1)))))))
a(b(b(a(a(x1))))) → b(b(a(b(b(b(a(x1)))))))
a(b(b(b(a(x1))))) → b(b(b(b(b(b(b(x1)))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP

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

A(a(a(a(x1)))) → A(a(b(a(a(x1)))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(x1))))) → A(a(b(b(a(a(x1))))))
A(a(b(a(x1)))) → A(b(x1))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(a(a(a(a(x1))))) → A(b(a(a(a(x1)))))
A(a(a(b(a(x1))))) → A(b(x1))
A(a(a(a(a(x1))))) → A(a(b(a(a(a(x1))))))
A(a(a(b(a(x1))))) → A(a(b(x1)))
A(b(a(b(a(x1))))) → A(b(x1))
A(a(a(x1))) → A(b(a(x1)))
A(a(a(a(x1)))) → A(b(a(a(x1))))
A(a(a(a(a(x1))))) → A(a(a(b(a(a(a(x1)))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(a(a(b(a(x1))))) → A(a(b(b(a(a(b(x1)))))))

The TRS R consists of the following rules:

a(a(x1)) → b(x1)
a(a(a(x1))) → a(b(a(x1)))
a(b(a(x1))) → b(b(b(x1)))
a(a(a(a(x1)))) → a(a(b(a(a(x1)))))
a(a(b(a(x1)))) → a(b(b(a(b(x1)))))
a(b(a(a(x1)))) → b(a(b(b(a(x1)))))
a(b(b(a(x1)))) → b(b(b(b(b(x1)))))
a(a(a(a(a(x1))))) → a(a(a(b(a(a(a(x1)))))))
a(a(a(b(a(x1))))) → a(a(b(b(a(a(b(x1)))))))
a(a(b(a(a(x1))))) → a(b(a(b(a(b(a(x1)))))))
a(a(b(b(a(x1))))) → a(b(b(b(a(b(b(x1)))))))
a(b(a(a(a(x1))))) → b(a(a(b(b(a(a(x1)))))))
a(b(a(b(a(x1))))) → b(a(b(b(b(a(b(x1)))))))
a(b(b(a(a(x1))))) → b(b(a(b(b(b(a(x1)))))))
a(b(b(b(a(x1))))) → b(b(b(b(b(b(b(x1)))))))

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