Termination w.r.t. Q proof of /tmp/tmp6g4SfE/17.srs

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ 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.

We have reversed the following QTRS:
The set of rules R is

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)))))))

The set Q is empty.
We have obtained the following QTRS:

a(a(x)) → b(x)
a(a(a(x))) → a(b(a(x)))
a(b(a(x))) → b(b(b(x)))
a(a(a(a(x)))) → a(a(b(a(a(x)))))
a(b(a(a(x)))) → b(a(b(b(a(x)))))
a(a(b(a(x)))) → a(b(b(a(b(x)))))
a(b(b(a(x)))) → b(b(b(b(b(x)))))
a(a(a(a(a(x))))) → a(a(a(b(a(a(a(x)))))))
a(b(a(a(a(x))))) → b(a(a(b(b(a(a(x)))))))
a(a(b(a(a(x))))) → a(b(a(b(a(b(a(x)))))))
a(b(b(a(a(x))))) → b(b(a(b(b(b(a(x)))))))
a(a(a(b(a(x))))) → a(a(b(b(a(a(b(x)))))))
a(b(a(b(a(x))))) → b(a(b(b(b(a(b(x)))))))
a(a(b(b(a(x))))) → a(b(b(b(a(b(b(x)))))))
a(b(b(b(a(x))))) → b(b(b(b(b(b(b(x)))))))

The set Q is empty.

↳ QTRS

  ↳ QTRS Reverse

    ↳ QTRS

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

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

a(a(x)) → b(x)
a(a(a(x))) → a(b(a(x)))
a(b(a(x))) → b(b(b(x)))
a(a(a(a(x)))) → a(a(b(a(a(x)))))
a(b(a(a(x)))) → b(a(b(b(a(x)))))
a(a(b(a(x)))) → a(b(b(a(b(x)))))
a(b(b(a(x)))) → b(b(b(b(b(x)))))
a(a(a(a(a(x))))) → a(a(a(b(a(a(a(x)))))))
a(b(a(a(a(x))))) → b(a(a(b(b(a(a(x)))))))
a(a(b(a(a(x))))) → a(b(a(b(a(b(a(x)))))))
a(b(b(a(a(x))))) → b(b(a(b(b(b(a(x)))))))
a(a(a(b(a(x))))) → a(a(b(b(a(a(b(x)))))))
a(b(a(b(a(x))))) → b(a(b(b(b(a(b(x)))))))
a(a(b(b(a(x))))) → a(b(b(b(a(b(b(x)))))))
a(b(b(b(a(x))))) → b(b(b(b(b(b(b(x)))))))

Q is empty.

We have reversed the following QTRS:
The set of rules R is

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)))))))

The set Q is empty.
We have obtained the following QTRS:

a(a(x)) → b(x)
a(a(a(x))) → a(b(a(x)))
a(b(a(x))) → b(b(b(x)))
a(a(a(a(x)))) → a(a(b(a(a(x)))))
a(b(a(a(x)))) → b(a(b(b(a(x)))))
a(a(b(a(x)))) → a(b(b(a(b(x)))))
a(b(b(a(x)))) → b(b(b(b(b(x)))))
a(a(a(a(a(x))))) → a(a(a(b(a(a(a(x)))))))
a(b(a(a(a(x))))) → b(a(a(b(b(a(a(x)))))))
a(a(b(a(a(x))))) → a(b(a(b(a(b(a(x)))))))
a(b(b(a(a(x))))) → b(b(a(b(b(b(a(x)))))))
a(a(a(b(a(x))))) → a(a(b(b(a(a(b(x)))))))
a(b(a(b(a(x))))) → b(a(b(b(b(a(b(x)))))))
a(a(b(b(a(x))))) → a(b(b(b(a(b(b(x)))))))
a(b(b(b(a(x))))) → b(b(b(b(b(b(b(x)))))))

The set Q is empty.

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

    ↳ QTRS

  ↳ DependencyPairsProof

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

a(a(x)) → b(x)
a(a(a(x))) → a(b(a(x)))
a(b(a(x))) → b(b(b(x)))
a(a(a(a(x)))) → a(a(b(a(a(x)))))
a(b(a(a(x)))) → b(a(b(b(a(x)))))
a(a(b(a(x)))) → a(b(b(a(b(x)))))
a(b(b(a(x)))) → b(b(b(b(b(x)))))
a(a(a(a(a(x))))) → a(a(a(b(a(a(a(x)))))))
a(b(a(a(a(x))))) → b(a(a(b(b(a(a(x)))))))
a(a(b(a(a(x))))) → a(b(a(b(a(b(a(x)))))))
a(b(b(a(a(x))))) → b(b(a(b(b(b(a(x)))))))
a(a(a(b(a(x))))) → a(a(b(b(a(a(b(x)))))))
a(b(a(b(a(x))))) → b(a(b(b(b(a(b(x)))))))
a(a(b(b(a(x))))) → a(b(b(b(a(b(b(x)))))))
a(b(b(b(a(x))))) → b(b(b(b(b(b(b(x)))))))

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

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ 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

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial Order [21,25] with Interpretation:

POL( A(x₁) ) = max{0, x₁ - 1}

POL( a(x₁) ) = x₁ + 1

POL( b(x₁) ) = max{0, -1}

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

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

A(b(a(b(a(x1))))) → A(b(x1))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(a(b(a(x1)))) → A(b(x1))
A(b(a(a(a(x1))))) → A(a(b(b(a(a(x1))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(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.
By narrowing [15] the rule A(b(a(a(a(x1))))) → A(a(b(b(a(a(x1)))))) at position [0] we obtained the following new rules:

A(b(a(a(a(a(a(b(a(x0))))))))) → A(a(b(b(a(a(a(b(b(a(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(b(x0))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(x0))))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(a(x0)))))))))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(b(x0))))))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(x0))))))))))
A(b(a(a(a(a(b(a(a(x0))))))))) → A(a(b(b(a(a(b(a(b(a(b(a(x0))))))))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(a(b(b(x0)))))))))))
A(b(a(a(a(b(b(b(a(x0))))))))) → A(a(b(b(a(b(b(b(b(b(b(b(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(b(a(x0))))))))))
A(b(a(a(a(b(a(b(a(x0))))))))) → A(a(b(b(a(b(a(b(b(b(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(a(b(x0)))))))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(b(b(x0))))))))))
A(b(a(a(a(x0))))) → A(b(b(a(b(b(b(a(x0))))))))
A(b(a(a(a(b(b(a(a(x0))))))))) → A(a(b(b(a(b(b(a(b(b(b(a(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(a(b(a(x0)))))))))))
A(b(a(a(a(b(a(a(a(x0))))))))) → A(a(b(b(a(b(a(a(b(b(a(a(x0))))))))))))
A(b(a(a(a(y0))))) → A(b(b(b(b(b(a(y0)))))))
A(b(a(a(a(x0))))) → A(a(b(b(b(x0)))))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(x0))))))))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(a(x0)))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(a(x0)))))))
A(b(a(a(a(a(b(b(a(x0))))))))) → A(a(b(b(a(a(b(b(b(a(b(b(x0))))))))))))
A(b(a(a(a(a(a(a(a(x0))))))))) → A(a(b(b(a(a(a(a(b(a(a(a(x0))))))))))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(a(b(x0)))))))))

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

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

A(b(a(a(a(a(a(b(a(x0))))))))) → A(a(b(b(a(a(a(b(b(a(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(b(x0))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(x0))))))
A(a(b(a(x1)))) → A(b(x1))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(a(x0)))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(b(x0))))))))
A(b(a(a(a(a(b(a(a(x0))))))))) → A(a(b(b(a(a(b(a(b(a(b(a(x0))))))))))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(x0))))))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(a(b(b(x0)))))))))))
A(b(a(a(a(b(b(b(a(x0))))))))) → A(a(b(b(a(b(b(b(b(b(b(b(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(b(a(x0))))))))))
A(b(a(a(a(b(a(b(a(x0))))))))) → A(a(b(b(a(b(a(b(b(b(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(a(b(x0)))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(b(b(x0))))))))))
A(b(a(a(a(x0))))) → A(b(b(a(b(b(b(a(x0))))))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(b(b(a(a(x0))))))))) → A(a(b(b(a(b(b(a(b(b(b(a(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(a(b(a(x0)))))))))))
A(b(a(a(a(b(a(a(a(x0))))))))) → A(a(b(b(a(b(a(a(b(b(a(a(x0))))))))))))
A(b(a(a(a(y0))))) → A(b(b(b(b(b(a(y0)))))))
A(b(a(a(a(x0))))) → A(a(b(b(b(x0)))))
A(b(a(b(a(x1))))) → A(b(x1))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(x0))))))))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(a(x0)))))))))
A(b(a(a(a(a(b(b(a(x0))))))))) → A(a(b(b(a(a(b(b(b(a(b(b(x0))))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(a(x0)))))))
A(b(a(a(a(a(a(a(a(x0))))))))) → A(a(b(b(a(a(a(a(b(a(a(a(x0))))))))))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(a(b(x0)))))))))

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 2 less nodes.

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

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

A(b(a(a(a(a(a(b(a(x0))))))))) → A(a(b(b(a(a(a(b(b(a(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(b(x0))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(x0))))))
A(a(b(a(x1)))) → A(b(x1))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(a(x0)))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(b(x0))))))))
A(b(a(a(a(a(b(a(a(x0))))))))) → A(a(b(b(a(a(b(a(b(a(b(a(x0))))))))))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(x0))))))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(a(b(b(x0)))))))))))
A(b(a(a(a(b(b(b(a(x0))))))))) → A(a(b(b(a(b(b(b(b(b(b(b(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(b(a(x0))))))))))
A(b(a(a(a(b(a(b(a(x0))))))))) → A(a(b(b(a(b(a(b(b(b(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(a(b(x0)))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(b(b(x0))))))))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(b(b(a(a(x0))))))))) → A(a(b(b(a(b(b(a(b(b(b(a(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(a(b(a(x0)))))))))))
A(b(a(a(a(b(a(a(a(x0))))))))) → A(a(b(b(a(b(a(a(b(b(a(a(x0))))))))))))
A(b(a(a(a(x0))))) → A(a(b(b(b(x0)))))
A(b(a(b(a(x1))))) → A(b(x1))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(x0))))))))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(a(x0)))))))))
A(b(a(a(a(a(b(b(a(x0))))))))) → A(a(b(b(a(a(b(b(b(a(b(b(x0))))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(a(x0)))))))
A(b(a(a(a(a(a(a(a(x0))))))))) → A(a(b(b(a(a(a(a(b(a(a(a(x0))))))))))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(a(b(x0)))))))))

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.
By narrowing [15] the rule A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(b(x0)))))))) at position [0] we obtained the following new rules:

A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(b(b(b(b(b(b(b(x0)))))))))))
A(b(a(a(a(b(a(y0))))))) → A(b(b(b(b(b(b(b(b(y0)))))))))

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

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

A(b(a(a(a(a(a(b(a(x0))))))))) → A(a(b(b(a(a(a(b(b(a(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(b(x0))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(x0))))))
A(a(b(a(x1)))) → A(b(x1))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(a(x0)))))))))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(x0))))))))))
A(b(a(a(a(a(b(a(a(x0))))))))) → A(a(b(b(a(a(b(a(b(a(b(a(x0))))))))))))
A(b(a(a(a(b(b(b(a(x0))))))))) → A(a(b(b(a(b(b(b(b(b(b(b(x0))))))))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(a(b(b(x0)))))))))))
A(b(a(a(a(b(a(b(a(x0))))))))) → A(a(b(b(a(b(a(b(b(b(a(b(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(b(a(x0))))))))))
A(b(a(a(a(b(a(y0))))))) → A(b(b(b(b(b(b(b(b(y0)))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(a(b(x0)))))))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(b(b(x0))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(b(b(a(a(x0))))))))) → A(a(b(b(a(b(b(a(b(b(b(a(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(a(b(a(x0)))))))))))
A(b(a(a(a(b(a(a(a(x0))))))))) → A(a(b(b(a(b(a(a(b(b(a(a(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(b(b(b(b(b(b(b(x0)))))))))))
A(b(a(a(a(x0))))) → A(a(b(b(b(x0)))))
A(b(a(b(a(x1))))) → A(b(x1))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(x0))))))))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(a(x0)))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(a(x0)))))))
A(b(a(a(a(a(b(b(a(x0))))))))) → A(a(b(b(a(a(b(b(b(a(b(b(x0))))))))))))
A(b(a(a(a(a(a(a(a(x0))))))))) → A(a(b(b(a(a(a(a(b(a(a(a(x0))))))))))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(a(b(x0)))))))))

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 2 less nodes.

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

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

A(b(a(a(a(a(a(b(a(x0))))))))) → A(a(b(b(a(a(a(b(b(a(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(b(x0))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(x0))))))
A(a(b(a(x1)))) → A(b(x1))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(a(x0)))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(b(a(a(a(a(b(a(a(x0))))))))) → A(a(b(b(a(a(b(a(b(a(b(a(x0))))))))))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(x0))))))))))
A(b(a(a(a(b(b(b(a(x0))))))))) → A(a(b(b(a(b(b(b(b(b(b(b(x0))))))))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(a(b(b(x0)))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(b(a(x0))))))))))
A(b(a(a(a(b(a(b(a(x0))))))))) → A(a(b(b(a(b(a(b(b(b(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(a(b(x0)))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(b(b(x0))))))))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(b(b(a(a(x0))))))))) → A(a(b(b(a(b(b(a(b(b(b(a(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(a(b(a(x0)))))))))))
A(b(a(a(a(b(a(a(a(x0))))))))) → A(a(b(b(a(b(a(a(b(b(a(a(x0))))))))))))
A(b(a(a(a(x0))))) → A(a(b(b(b(x0)))))
A(b(a(b(a(x1))))) → A(b(x1))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(x0))))))))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(a(x0)))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(a(x0)))))))
A(b(a(a(a(a(b(b(a(x0))))))))) → A(a(b(b(a(a(b(b(b(a(b(b(x0))))))))))))
A(b(a(a(a(a(a(a(a(x0))))))))) → A(a(b(b(a(a(a(a(b(a(a(a(x0))))))))))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(a(b(x0)))))))))

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.
By narrowing [15] the rule A(b(a(a(a(b(b(b(a(x0))))))))) → A(a(b(b(a(b(b(b(b(b(b(b(x0)))))))))))) at position [0] we obtained the following new rules:

A(b(a(a(a(b(b(b(a(y0))))))))) → A(b(b(b(b(b(b(b(b(b(b(b(b(y0)))))))))))))

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

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

A(b(a(a(a(a(a(b(a(x0))))))))) → A(a(b(b(a(a(a(b(b(a(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(b(x0))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(x0))))))
A(a(b(a(x1)))) → A(b(x1))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(a(x0)))))))))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(x0))))))))))
A(b(a(a(a(a(b(a(a(x0))))))))) → A(a(b(b(a(a(b(a(b(a(b(a(x0))))))))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(a(b(b(x0)))))))))))
A(b(a(a(a(b(a(b(a(x0))))))))) → A(a(b(b(a(b(a(b(b(b(a(b(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(b(a(x0))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(a(b(x0)))))))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(b(b(x0))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(b(a(a(a(b(b(b(a(y0))))))))) → A(b(b(b(b(b(b(b(b(b(b(b(b(y0)))))))))))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(b(b(a(a(x0))))))))) → A(a(b(b(a(b(b(a(b(b(b(a(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(a(b(a(x0)))))))))))
A(b(a(a(a(b(a(a(a(x0))))))))) → A(a(b(b(a(b(a(a(b(b(a(a(x0))))))))))))
A(b(a(a(a(x0))))) → A(a(b(b(b(x0)))))
A(b(a(b(a(x1))))) → A(b(x1))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(x0))))))))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(a(x0)))))))))
A(b(a(a(a(a(b(b(a(x0))))))))) → A(a(b(b(a(a(b(b(b(a(b(b(x0))))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(a(x0)))))))
A(b(a(a(a(a(a(a(a(x0))))))))) → A(a(b(b(a(a(a(a(b(a(a(a(x0))))))))))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(a(b(x0)))))))))

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 1 less node.

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

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

A(b(a(a(a(a(a(b(a(x0))))))))) → A(a(b(b(a(a(a(b(b(a(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(b(x0))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(x0))))))
A(a(b(a(x1)))) → A(b(x1))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(a(x0)))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(b(a(a(a(a(b(a(a(x0))))))))) → A(a(b(b(a(a(b(a(b(a(b(a(x0))))))))))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(x0))))))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(a(b(b(x0)))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(b(a(x0))))))))))
A(b(a(a(a(b(a(b(a(x0))))))))) → A(a(b(b(a(b(a(b(b(b(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(a(b(x0)))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(b(b(x0))))))))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(b(b(a(a(x0))))))))) → A(a(b(b(a(b(b(a(b(b(b(a(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(a(b(a(x0)))))))))))
A(b(a(a(a(b(a(a(a(x0))))))))) → A(a(b(b(a(b(a(a(b(b(a(a(x0))))))))))))
A(b(a(a(a(x0))))) → A(a(b(b(b(x0)))))
A(b(a(b(a(x1))))) → A(b(x1))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(x0))))))))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(a(x0)))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(a(x0)))))))
A(b(a(a(a(a(b(b(a(x0))))))))) → A(a(b(b(a(a(b(b(b(a(b(b(x0))))))))))))
A(b(a(a(a(a(a(a(a(x0))))))))) → A(a(b(b(a(a(a(a(b(a(a(a(x0))))))))))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(a(b(x0)))))))))

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.
By narrowing [15] the rule A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(b(b(x0)))))))))) at position [0] we obtained the following new rules:

A(b(a(a(a(b(b(a(y0)))))))) → A(b(b(b(b(b(b(b(b(b(b(y0)))))))))))

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

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

A(b(a(a(a(a(a(b(a(x0))))))))) → A(a(b(b(a(a(a(b(b(a(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(b(x0))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(x0))))))
A(a(b(a(x1)))) → A(b(x1))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(a(x0)))))))))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(x0))))))))))
A(b(a(a(a(a(b(a(a(x0))))))))) → A(a(b(b(a(a(b(a(b(a(b(a(x0))))))))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(a(b(b(x0)))))))))))
A(b(a(a(a(b(a(b(a(x0))))))))) → A(a(b(b(a(b(a(b(b(b(a(b(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(b(a(x0))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(a(b(x0)))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(b(b(a(a(x0))))))))) → A(a(b(b(a(b(b(a(b(b(b(a(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(a(b(a(x0)))))))))))
A(b(a(a(a(b(a(a(a(x0))))))))) → A(a(b(b(a(b(a(a(b(b(a(a(x0))))))))))))
A(b(a(a(a(b(b(a(y0)))))))) → A(b(b(b(b(b(b(b(b(b(b(y0)))))))))))
A(b(a(a(a(x0))))) → A(a(b(b(b(x0)))))
A(b(a(b(a(x1))))) → A(b(x1))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(x0))))))))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(a(x0)))))))))
A(b(a(a(a(a(b(b(a(x0))))))))) → A(a(b(b(a(a(b(b(b(a(b(b(x0))))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(a(x0)))))))
A(b(a(a(a(a(a(a(a(x0))))))))) → A(a(b(b(a(a(a(a(b(a(a(a(x0))))))))))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(a(b(x0)))))))))

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 1 less node.

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

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

A(b(a(a(a(a(a(b(a(x0))))))))) → A(a(b(b(a(a(a(b(b(a(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(b(x0))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(x0))))))
A(a(b(a(x1)))) → A(b(x1))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(a(x0)))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(b(a(a(a(a(b(a(a(x0))))))))) → A(a(b(b(a(a(b(a(b(a(b(a(x0))))))))))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(x0))))))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(a(b(b(x0)))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(b(a(x0))))))))))
A(b(a(a(a(b(a(b(a(x0))))))))) → A(a(b(b(a(b(a(b(b(b(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(a(b(x0)))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(b(b(a(a(x0))))))))) → A(a(b(b(a(b(b(a(b(b(b(a(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(a(b(a(x0)))))))))))
A(b(a(a(a(b(a(a(a(x0))))))))) → A(a(b(b(a(b(a(a(b(b(a(a(x0))))))))))))
A(b(a(a(a(x0))))) → A(a(b(b(b(x0)))))
A(b(a(b(a(x1))))) → A(b(x1))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(x0))))))))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(a(x0)))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(a(x0)))))))
A(b(a(a(a(a(b(b(a(x0))))))))) → A(a(b(b(a(a(b(b(b(a(b(b(x0))))))))))))
A(b(a(a(a(a(a(a(a(x0))))))))) → A(a(b(b(a(a(a(a(b(a(a(a(x0))))))))))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(a(b(x0)))))))))

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.
By narrowing [15] the rule A(b(a(a(a(x0))))) → A(a(b(b(b(x0))))) at position [0] we obtained the following new rules:

A(b(a(a(a(a(x0)))))) → A(b(b(b(b(b(b(b(x0))))))))

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

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

A(b(a(a(a(a(a(b(a(x0))))))))) → A(a(b(b(a(a(a(b(b(a(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(b(x0))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(x0))))))
A(a(b(a(x1)))) → A(b(x1))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(a(x0)))))))))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(x0))))))))))
A(b(a(a(a(a(b(a(a(x0))))))))) → A(a(b(b(a(a(b(a(b(a(b(a(x0))))))))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(a(b(b(x0)))))))))))
A(b(a(a(a(b(a(b(a(x0))))))))) → A(a(b(b(a(b(a(b(b(b(a(b(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(b(a(x0))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(a(b(x0)))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(b(b(a(a(x0))))))))) → A(a(b(b(a(b(b(a(b(b(b(a(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(a(b(a(x0)))))))))))
A(b(a(a(a(b(a(a(a(x0))))))))) → A(a(b(b(a(b(a(a(b(b(a(a(x0))))))))))))
A(b(a(b(a(x1))))) → A(b(x1))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(x0))))))))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(a(x0)))))))))
A(b(a(a(a(a(b(b(a(x0))))))))) → A(a(b(b(a(a(b(b(b(a(b(b(x0))))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(a(x0)))))))
A(b(a(a(a(a(a(a(a(x0))))))))) → A(a(b(b(a(a(a(a(b(a(a(a(x0))))))))))))
A(b(a(a(a(a(x0)))))) → A(b(b(b(b(b(b(b(x0))))))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(a(b(x0)))))))))

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 1 less node.

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ SemLabProof

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

A(b(a(a(a(a(a(b(a(x0))))))))) → A(a(b(b(a(a(a(b(b(a(a(b(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(b(x0))))))))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(b(b(a(a(x0))))))))) → A(a(b(b(a(b(b(a(b(b(b(a(x0))))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(x0))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(a(b(a(x0)))))))))))
A(b(a(a(a(b(a(a(a(x0))))))))) → A(a(b(b(a(b(a(a(b(b(a(a(x0))))))))))))
A(a(b(a(x1)))) → A(b(x1))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(a(x0)))))))))))
A(b(a(a(a(a(a(a(x0)))))))) → A(a(b(b(a(a(a(b(a(a(x0))))))))))
A(b(a(a(a(a(b(a(a(x0))))))))) → A(a(b(b(a(a(b(a(b(a(b(a(x0))))))))))))
A(b(a(b(a(x1))))) → A(b(x1))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(x0))))))))
A(b(a(a(a(a(a(x0))))))) → A(a(b(b(a(a(b(a(a(x0)))))))))
A(b(a(a(a(a(b(b(a(x0))))))))) → A(a(b(b(a(a(b(b(b(a(b(b(x0))))))))))))
A(b(a(a(a(a(x0)))))) → A(a(b(b(a(b(a(x0)))))))
A(b(a(a(a(b(b(a(x0)))))))) → A(a(b(b(a(b(b(b(a(b(b(x0)))))))))))
A(b(a(a(a(b(a(b(a(x0))))))))) → A(a(b(b(a(b(a(b(b(b(a(b(x0))))))))))))
A(b(a(a(a(b(a(a(x0)))))))) → A(a(b(b(a(b(a(b(b(a(x0))))))))))
A(b(a(a(a(a(a(a(a(x0))))))))) → A(a(b(b(a(a(a(a(b(a(a(a(x0))))))))))))
A(b(a(a(a(a(b(a(x0)))))))) → A(a(b(b(a(a(b(b(a(a(b(x0)))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(b(a(a(a(b(a(x0))))))) → A(a(b(b(a(b(b(a(b(x0)))))))))

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.
We found the following quasi-model for the rules of the TRS R. Interpretation over the domain with elements from 0 to 1.a: 1
A: 0
b: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:

A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.0(b.1(a.0(x1))))) → A.0(b.0(x1))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))))
A.0(b.1(a.0(b.1(a.1(x1))))) → A.0(b.1(x1))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(x0)))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.1(x0))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.0(x0))))))))
A.1(a.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.0(b.1(a.0(b.1(a.1(x1)))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.0(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.0(x0))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0)))))))))))
A.1(a.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.1(x1)))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.1(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.1(x0))))))))))
A.1(a.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(b.1(a.0(b.1(a.0(x1)))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(x0))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.1(x0))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.1(x0))))))))
A.1(a.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.0(b.1(a.1(x1)))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.1(x0))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.1(x0)))))))
A.1(a.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(x1)))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(x0))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(x0))))))))))))
A.1(a.0(b.1(a.1(x1)))) → A.0(b.1(x1))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.1(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.0(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(x0))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.1(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(x0)))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(x0))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.0(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.1(x0)))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0)))))))))))
A.1(a.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(b.1(a.0(x1)))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.1(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.0(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.1(x0))))))))))))
A.1(a.0(b.1(a.0(x1)))) → A.0(b.0(x1))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(x0))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.1(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.0(x0))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))))

The TRS R consists of the following rules:

a.0(b.0(b.0(b.1(a.0(x1))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(x1)))))))
a.0(b.1(a.0(x1))) → b.0(b.0(b.0(x1)))
a.1(a.1(a.1(x1))) → a.0(b.1(a.1(x1)))
a.1(a.1(a.1(a.1(x1)))) → a.1(a.0(b.1(a.1(a.1(x1)))))
b.1(x0) → b.0(x0)
a.1(a.1(a.0(x1))) → a.0(b.1(a.0(x1)))
a.0(b.1(a.1(a.0(x1)))) → b.1(a.0(b.0(b.1(a.0(x1)))))
a.0(b.0(b.1(a.1(a.0(x1))))) → b.0(b.1(a.0(b.0(b.0(b.1(a.0(x1)))))))
a.1(x0) → a.0(x0)
a.1(a.1(x1)) → b.1(x1)
a.1(a.1(a.0(b.1(a.1(x1))))) → a.1(a.0(b.0(b.1(a.1(a.0(b.1(x1)))))))
a.0(b.1(a.1(a.1(a.1(x1))))) → b.1(a.1(a.0(b.0(b.1(a.1(a.1(x1)))))))
a.1(a.1(a.1(a.1(a.1(x1))))) → a.1(a.1(a.0(b.1(a.1(a.1(a.1(x1)))))))
a.0(b.1(a.1(a.1(x1)))) → b.1(a.0(b.0(b.1(a.1(x1)))))
a.1(a.1(a.1(a.1(a.0(x1))))) → a.1(a.1(a.0(b.1(a.1(a.1(a.0(x1)))))))
a.1(a.0(b.1(a.1(x1)))) → a.0(b.0(b.1(a.0(b.1(x1)))))
a.1(a.0(x1)) → b.0(x1)
a.1(a.0(b.1(a.1(a.0(x1))))) → a.0(b.1(a.0(b.1(a.0(b.1(a.0(x1)))))))
a.0(b.1(a.1(a.1(a.0(x1))))) → b.1(a.1(a.0(b.0(b.1(a.1(a.0(x1)))))))
a.1(a.0(b.1(a.0(x1)))) → a.0(b.0(b.1(a.0(b.0(x1)))))
a.0(b.1(a.0(b.1(a.1(x1))))) → b.1(a.0(b.0(b.0(b.1(a.0(b.1(x1)))))))
a.0(b.1(a.0(b.1(a.0(x1))))) → b.1(a.0(b.0(b.0(b.1(a.0(b.0(x1)))))))
a.0(b.0(b.0(b.1(a.1(x1))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.1(x1)))))))
a.0(b.0(b.1(a.1(a.1(x1))))) → b.0(b.1(a.0(b.0(b.0(b.1(a.1(x1)))))))
a.1(a.1(a.0(b.1(a.0(x1))))) → a.1(a.0(b.0(b.1(a.1(a.0(b.0(x1)))))))
a.0(b.0(b.1(a.1(x1)))) → b.0(b.0(b.0(b.0(b.1(x1)))))
a.1(a.0(b.0(b.1(a.0(x1))))) → a.0(b.0(b.0(b.1(a.0(b.0(b.0(x1)))))))
a.1(a.1(a.1(a.0(x1)))) → a.1(a.0(b.1(a.1(a.0(x1)))))
a.0(b.1(a.1(x1))) → b.0(b.0(b.1(x1)))
a.0(b.0(b.1(a.0(x1)))) → b.0(b.0(b.0(b.0(b.0(x1)))))
a.1(a.0(b.0(b.1(a.1(x1))))) → a.0(b.0(b.0(b.1(a.0(b.0(b.1(x1)))))))
a.1(a.0(b.1(a.1(a.1(x1))))) → a.0(b.1(a.0(b.1(a.0(b.1(a.1(x1)))))))

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

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ SemLabProof

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

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

A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.0(b.1(a.0(x1))))) → A.0(b.0(x1))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))))
A.0(b.1(a.0(b.1(a.1(x1))))) → A.0(b.1(x1))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(x0)))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.1(x0))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.0(x0))))))))
A.1(a.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.0(b.1(a.0(b.1(a.1(x1)))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.0(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.0(x0))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0)))))))))))
A.1(a.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.1(x1)))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.1(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.1(x0))))))))))
A.1(a.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(b.1(a.0(b.1(a.0(x1)))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(x0))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.1(x0))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.1(x0))))))))
A.1(a.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.0(b.1(a.1(x1)))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.1(x0))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.1(x0)))))))
A.1(a.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(x1)))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(x0))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(x0))))))))))))
A.1(a.0(b.1(a.1(x1)))) → A.0(b.1(x1))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.1(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.0(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(x0))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.1(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(x0)))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(x0))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.0(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.1(x0)))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0)))))))))))
A.1(a.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(b.1(a.0(x1)))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.1(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.0(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.1(x0))))))))))))
A.1(a.0(b.1(a.0(x1)))) → A.0(b.0(x1))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(x0))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.1(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.0(x0))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))))

The TRS R consists of the following rules:

a.0(b.0(b.0(b.1(a.0(x1))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(x1)))))))
a.0(b.1(a.0(x1))) → b.0(b.0(b.0(x1)))
a.1(a.1(a.1(x1))) → a.0(b.1(a.1(x1)))
a.1(a.1(a.1(a.1(x1)))) → a.1(a.0(b.1(a.1(a.1(x1)))))
b.1(x0) → b.0(x0)
a.1(a.1(a.0(x1))) → a.0(b.1(a.0(x1)))
a.0(b.1(a.1(a.0(x1)))) → b.1(a.0(b.0(b.1(a.0(x1)))))
a.0(b.0(b.1(a.1(a.0(x1))))) → b.0(b.1(a.0(b.0(b.0(b.1(a.0(x1)))))))
a.1(x0) → a.0(x0)
a.1(a.1(x1)) → b.1(x1)
a.1(a.1(a.0(b.1(a.1(x1))))) → a.1(a.0(b.0(b.1(a.1(a.0(b.1(x1)))))))
a.0(b.1(a.1(a.1(a.1(x1))))) → b.1(a.1(a.0(b.0(b.1(a.1(a.1(x1)))))))
a.1(a.1(a.1(a.1(a.1(x1))))) → a.1(a.1(a.0(b.1(a.1(a.1(a.1(x1)))))))
a.0(b.1(a.1(a.1(x1)))) → b.1(a.0(b.0(b.1(a.1(x1)))))
a.1(a.1(a.1(a.1(a.0(x1))))) → a.1(a.1(a.0(b.1(a.1(a.1(a.0(x1)))))))
a.1(a.0(b.1(a.1(x1)))) → a.0(b.0(b.1(a.0(b.1(x1)))))
a.1(a.0(x1)) → b.0(x1)
a.1(a.0(b.1(a.1(a.0(x1))))) → a.0(b.1(a.0(b.1(a.0(b.1(a.0(x1)))))))
a.0(b.1(a.1(a.1(a.0(x1))))) → b.1(a.1(a.0(b.0(b.1(a.1(a.0(x1)))))))
a.1(a.0(b.1(a.0(x1)))) → a.0(b.0(b.1(a.0(b.0(x1)))))
a.0(b.1(a.0(b.1(a.1(x1))))) → b.1(a.0(b.0(b.0(b.1(a.0(b.1(x1)))))))
a.0(b.1(a.0(b.1(a.0(x1))))) → b.1(a.0(b.0(b.0(b.1(a.0(b.0(x1)))))))
a.0(b.0(b.0(b.1(a.1(x1))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.1(x1)))))))
a.0(b.0(b.1(a.1(a.1(x1))))) → b.0(b.1(a.0(b.0(b.0(b.1(a.1(x1)))))))
a.1(a.1(a.0(b.1(a.0(x1))))) → a.1(a.0(b.0(b.1(a.1(a.0(b.0(x1)))))))
a.0(b.0(b.1(a.1(x1)))) → b.0(b.0(b.0(b.0(b.1(x1)))))
a.1(a.0(b.0(b.1(a.0(x1))))) → a.0(b.0(b.0(b.1(a.0(b.0(b.0(x1)))))))
a.1(a.1(a.1(a.0(x1)))) → a.1(a.0(b.1(a.1(a.0(x1)))))
a.0(b.1(a.1(x1))) → b.0(b.0(b.1(x1)))
a.0(b.0(b.1(a.0(x1)))) → b.0(b.0(b.0(b.0(b.0(x1)))))
a.1(a.0(b.0(b.1(a.1(x1))))) → a.0(b.0(b.0(b.1(a.0(b.0(b.1(x1)))))))
a.1(a.0(b.1(a.1(a.1(x1))))) → a.0(b.1(a.0(b.1(a.0(b.1(a.1(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 2 less nodes.

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ SemLabProof

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ RuleRemovalProof

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

A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(x0))))))))))
A.0(b.1(a.0(b.1(a.1(x1))))) → A.0(b.1(x1))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(x0)))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.1(x0))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.0(x0))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.0(x0)))))))))
A.1(a.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.0(b.1(a.0(b.1(a.1(x1)))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.0(x0))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0)))))))))))
A.1(a.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.1(x1)))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.1(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0))))))))))))
A.1(a.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(b.1(a.0(b.1(a.0(x1)))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(x0))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.1(x0))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.1(x0))))))))
A.1(a.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.0(b.1(a.1(x1)))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.1(x0))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.1(x0)))))))
A.1(a.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(x1)))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(x0))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0)))))))))))
A.1(a.0(b.1(a.1(x1)))) → A.0(b.1(x1))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.1(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.0(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(x0))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.1(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(x0)))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(x0))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.0(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.1(x0)))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.1(x0)))))))))
A.1(a.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(b.1(a.0(x1)))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.0(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(x0))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.1(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.0(x0))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))))

The TRS R consists of the following rules:

a.0(b.0(b.0(b.1(a.0(x1))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(x1)))))))
a.0(b.1(a.0(x1))) → b.0(b.0(b.0(x1)))
a.1(a.1(a.1(x1))) → a.0(b.1(a.1(x1)))
a.1(a.1(a.1(a.1(x1)))) → a.1(a.0(b.1(a.1(a.1(x1)))))
b.1(x0) → b.0(x0)
a.1(a.1(a.0(x1))) → a.0(b.1(a.0(x1)))
a.0(b.1(a.1(a.0(x1)))) → b.1(a.0(b.0(b.1(a.0(x1)))))
a.0(b.0(b.1(a.1(a.0(x1))))) → b.0(b.1(a.0(b.0(b.0(b.1(a.0(x1)))))))
a.1(x0) → a.0(x0)
a.1(a.1(x1)) → b.1(x1)
a.1(a.1(a.0(b.1(a.1(x1))))) → a.1(a.0(b.0(b.1(a.1(a.0(b.1(x1)))))))
a.0(b.1(a.1(a.1(a.1(x1))))) → b.1(a.1(a.0(b.0(b.1(a.1(a.1(x1)))))))
a.1(a.1(a.1(a.1(a.1(x1))))) → a.1(a.1(a.0(b.1(a.1(a.1(a.1(x1)))))))
a.0(b.1(a.1(a.1(x1)))) → b.1(a.0(b.0(b.1(a.1(x1)))))
a.1(a.1(a.1(a.1(a.0(x1))))) → a.1(a.1(a.0(b.1(a.1(a.1(a.0(x1)))))))
a.1(a.0(b.1(a.1(x1)))) → a.0(b.0(b.1(a.0(b.1(x1)))))
a.1(a.0(x1)) → b.0(x1)
a.1(a.0(b.1(a.1(a.0(x1))))) → a.0(b.1(a.0(b.1(a.0(b.1(a.0(x1)))))))
a.0(b.1(a.1(a.1(a.0(x1))))) → b.1(a.1(a.0(b.0(b.1(a.1(a.0(x1)))))))
a.1(a.0(b.1(a.0(x1)))) → a.0(b.0(b.1(a.0(b.0(x1)))))
a.0(b.1(a.0(b.1(a.1(x1))))) → b.1(a.0(b.0(b.0(b.1(a.0(b.1(x1)))))))
a.0(b.1(a.0(b.1(a.0(x1))))) → b.1(a.0(b.0(b.0(b.1(a.0(b.0(x1)))))))
a.0(b.0(b.0(b.1(a.1(x1))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.1(x1)))))))
a.0(b.0(b.1(a.1(a.1(x1))))) → b.0(b.1(a.0(b.0(b.0(b.1(a.1(x1)))))))
a.1(a.1(a.0(b.1(a.0(x1))))) → a.1(a.0(b.0(b.1(a.1(a.0(b.0(x1)))))))
a.0(b.0(b.1(a.1(x1)))) → b.0(b.0(b.0(b.0(b.1(x1)))))
a.1(a.0(b.0(b.1(a.0(x1))))) → a.0(b.0(b.0(b.1(a.0(b.0(b.0(x1)))))))
a.1(a.1(a.1(a.0(x1)))) → a.1(a.0(b.1(a.1(a.0(x1)))))
a.0(b.1(a.1(x1))) → b.0(b.0(b.1(x1)))
a.0(b.0(b.1(a.0(x1)))) → b.0(b.0(b.0(b.0(b.0(x1)))))
a.1(a.0(b.0(b.1(a.1(x1))))) → a.0(b.0(b.0(b.1(a.0(b.0(b.1(x1)))))))
a.1(a.0(b.1(a.1(a.1(x1))))) → a.0(b.1(a.0(b.1(a.0(b.1(a.1(x1)))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(x0))))))))))
A.0(b.1(a.0(b.1(a.1(x1))))) → A.0(b.1(x1))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(x0)))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.1(x0))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.0(x0))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.0(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.0(x0))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.1(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(b.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(x0))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.1(x0))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.1(x0))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.1(x0))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.1(a.0(b.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.1(x0)))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(x0))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.1(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.0(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.0(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(x0))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.1(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(x0)))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.1(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(x0))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.0(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(x0)))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.1(x0)))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(x0))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.1(a.1(a.1(x0)))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.1(a.0(b.0(b.1(a.0(b.0(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(x0)))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(x0)))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(a.0(x0))))))))) → A.0(a.0(b.0(b.1(a.1(a.1(a.1(a.0(b.1(a.1(a.1(a.0(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.0(b.1(a.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(b.1(a.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0)))))))) → A.0(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(x0))))))))))
A.0(b.1(a.1(a.1(a.0(b.1(a.1(x0))))))) → A.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.1(x0)))))))))
A.0(b.1(a.1(a.1(a.1(a.0(x0)))))) → A.0(a.0(b.0(b.1(a.0(b.0(x0))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.1(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.1(x0))))))))))
A.0(b.1(a.1(a.1(a.1(a.0(b.0(b.1(a.1(x0))))))))) → A.1(a.0(b.0(b.1(a.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(x0))))))))))))
A.0(b.1(a.1(a.1(a.1(a.1(a.1(a.0(x0)))))))) → A.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(a.1(a.0(x0))))))))))

Strictly oriented rules of the TRS R:

a.1(a.1(a.1(x1))) → a.0(b.1(a.1(x1)))
a.1(a.1(a.1(a.1(x1)))) → a.1(a.0(b.1(a.1(a.1(x1)))))
a.1(a.1(a.0(x1))) → a.0(b.1(a.0(x1)))
a.0(b.1(a.1(a.0(x1)))) → b.1(a.0(b.0(b.1(a.0(x1)))))
a.0(b.0(b.1(a.1(a.0(x1))))) → b.0(b.1(a.0(b.0(b.0(b.1(a.0(x1)))))))
a.1(x0) → a.0(x0)
a.1(a.1(x1)) → b.1(x1)
a.1(a.1(a.0(b.1(a.1(x1))))) → a.1(a.0(b.0(b.1(a.1(a.0(b.1(x1)))))))
a.0(b.1(a.1(a.1(x1)))) → b.1(a.0(b.0(b.1(a.1(x1)))))
a.1(a.0(b.1(a.1(x1)))) → a.0(b.0(b.1(a.0(b.1(x1)))))
a.1(a.0(x1)) → b.0(x1)
a.1(a.0(b.1(a.1(a.0(x1))))) → a.0(b.1(a.0(b.1(a.0(b.1(a.0(x1)))))))
a.1(a.0(b.1(a.0(x1)))) → a.0(b.0(b.1(a.0(b.0(x1)))))
a.0(b.1(a.0(b.1(a.1(x1))))) → b.1(a.0(b.0(b.0(b.1(a.0(b.1(x1)))))))
a.0(b.0(b.0(b.1(a.1(x1))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.1(x1)))))))
a.0(b.0(b.1(a.1(a.1(x1))))) → b.0(b.1(a.0(b.0(b.0(b.1(a.1(x1)))))))
a.0(b.0(b.1(a.1(x1)))) → b.0(b.0(b.0(b.0(b.1(x1)))))
a.1(a.0(b.0(b.1(a.0(x1))))) → a.0(b.0(b.0(b.1(a.0(b.0(b.0(x1)))))))
a.1(a.1(a.1(a.0(x1)))) → a.1(a.0(b.1(a.1(a.0(x1)))))
a.0(b.1(a.1(x1))) → b.0(b.0(b.1(x1)))
a.1(a.0(b.0(b.1(a.1(x1))))) → a.0(b.0(b.0(b.1(a.0(b.0(b.1(x1)))))))
a.1(a.0(b.1(a.1(a.1(x1))))) → a.0(b.1(a.0(b.1(a.0(b.1(a.1(x1)))))))

Used ordering: POLO with Polynomial interpretation [25]:

POL(A.0(x₁)) = 1 + x₁
POL(A.1(x₁)) = x₁
POL(a.0(x₁)) = x₁
POL(a.1(x₁)) = 1 + x₁
POL(b.0(x₁)) = x₁
POL(b.1(x₁)) = x₁

↳ QTRS

  ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ SemLabProof

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ RuleRemovalProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

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

A.1(a.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.0(b.1(a.0(b.1(a.1(x1)))))))
A.1(a.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.0(b.1(a.1(x1)))))
A.1(a.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(x1)))
A.1(a.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.1(x1)))
A.1(a.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(b.1(a.0(b.1(a.0(x1)))))))
A.1(a.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(b.1(a.0(x1)))))
A.1(a.0(b.1(a.1(x1)))) → A.0(b.1(x1))

The TRS R consists of the following rules:

a.0(b.0(b.0(b.1(a.0(x1))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(x1)))))))
a.0(b.1(a.0(x1))) → b.0(b.0(b.0(x1)))
b.1(x0) → b.0(x0)
a.0(b.1(a.1(a.1(a.1(x1))))) → b.1(a.1(a.0(b.0(b.1(a.1(a.1(x1)))))))
a.1(a.1(a.1(a.1(a.1(x1))))) → a.1(a.1(a.0(b.1(a.1(a.1(a.1(x1)))))))
a.1(a.1(a.1(a.1(a.0(x1))))) → a.1(a.1(a.0(b.1(a.1(a.1(a.0(x1)))))))
a.0(b.1(a.1(a.1(a.0(x1))))) → b.1(a.1(a.0(b.0(b.1(a.1(a.0(x1)))))))
a.0(b.1(a.0(b.1(a.0(x1))))) → b.1(a.0(b.0(b.0(b.1(a.0(b.0(x1)))))))
a.1(a.1(a.0(b.1(a.0(x1))))) → a.1(a.0(b.0(b.1(a.1(a.0(b.0(x1)))))))
a.0(b.0(b.1(a.0(x1)))) → b.0(b.0(b.0(b.0(b.0(x1)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 7 less nodes.