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

Q is empty.

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

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

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

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

Q is empty.

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

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

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

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

The TRS R consists of the following rules:

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

The TRS R consists of the following rules:

a(a(x1)) → b(a(b(x1)))
a(a(a(x1))) → a(b(a(b(a(x1)))))
a(b(a(x1))) → b(b(a(b(b(x1)))))
a(a(a(a(x1)))) → a(a(b(a(b(a(a(x1)))))))
a(a(b(a(x1)))) → a(b(b(a(b(a(b(x1)))))))
a(b(a(a(x1)))) → b(a(b(a(b(b(a(x1)))))))
a(b(b(a(x1)))) → b(b(b(a(b(b(b(x1)))))))
a(a(a(a(a(x1))))) → a(a(a(b(a(b(a(a(a(x1)))))))))
a(a(a(b(a(x1))))) → a(a(b(b(a(b(a(a(b(x1)))))))))
a(a(b(a(a(x1))))) → a(b(a(b(a(b(a(b(a(x1)))))))))
a(a(b(b(a(x1))))) → a(b(b(b(a(b(a(b(b(x1)))))))))
a(b(a(a(a(x1))))) → b(a(a(b(a(b(b(a(a(x1)))))))))
a(b(a(b(a(x1))))) → b(a(b(b(a(b(b(a(b(x1)))))))))
a(b(b(a(a(x1))))) → b(b(a(b(a(b(b(b(a(x1)))))))))
a(b(b(b(a(x1))))) → b(b(b(b(a(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 4 less nodes.

↳ QTRS
  ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing

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

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

The TRS R consists of the following rules:

a(a(x1)) → b(a(b(x1)))
a(a(a(x1))) → a(b(a(b(a(x1)))))
a(b(a(x1))) → b(b(a(b(b(x1)))))
a(a(a(a(x1)))) → a(a(b(a(b(a(a(x1)))))))
a(a(b(a(x1)))) → a(b(b(a(b(a(b(x1)))))))
a(b(a(a(x1)))) → b(a(b(a(b(b(a(x1)))))))
a(b(b(a(x1)))) → b(b(b(a(b(b(b(x1)))))))
a(a(a(a(a(x1))))) → a(a(a(b(a(b(a(a(a(x1)))))))))
a(a(a(b(a(x1))))) → a(a(b(b(a(b(a(a(b(x1)))))))))
a(a(b(a(a(x1))))) → a(b(a(b(a(b(a(b(a(x1)))))))))
a(a(b(b(a(x1))))) → a(b(b(b(a(b(a(b(b(x1)))))))))
a(b(a(a(a(x1))))) → b(a(a(b(a(b(b(a(a(x1)))))))))
a(b(a(b(a(x1))))) → b(a(b(b(a(b(b(a(b(x1)))))))))
a(b(b(a(a(x1))))) → b(b(a(b(a(b(b(b(a(x1)))))))))
a(b(b(b(a(x1))))) → b(b(b(b(a(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(b(a(x1))))) → A(b(b(a(b(x1))))) at position [0,0,0] we obtained the following new rules:

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



↳ QTRS
  ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ DependencyGraphProof

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

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

The TRS R consists of the following rules:

a(a(x1)) → b(a(b(x1)))
a(a(a(x1))) → a(b(a(b(a(x1)))))
a(b(a(x1))) → b(b(a(b(b(x1)))))
a(a(a(a(x1)))) → a(a(b(a(b(a(a(x1)))))))
a(a(b(a(x1)))) → a(b(b(a(b(a(b(x1)))))))
a(b(a(a(x1)))) → b(a(b(a(b(b(a(x1)))))))
a(b(b(a(x1)))) → b(b(b(a(b(b(b(x1)))))))
a(a(a(a(a(x1))))) → a(a(a(b(a(b(a(a(a(x1)))))))))
a(a(a(b(a(x1))))) → a(a(b(b(a(b(a(a(b(x1)))))))))
a(a(b(a(a(x1))))) → a(b(a(b(a(b(a(b(a(x1)))))))))
a(a(b(b(a(x1))))) → a(b(b(b(a(b(a(b(b(x1)))))))))
a(b(a(a(a(x1))))) → b(a(a(b(a(b(b(a(a(x1)))))))))
a(b(a(b(a(x1))))) → b(a(b(b(a(b(b(a(b(x1)))))))))
a(b(b(a(a(x1))))) → b(b(a(b(a(b(b(b(a(x1)))))))))
a(b(b(b(a(x1))))) → b(b(b(b(a(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 7 less nodes.

↳ QTRS
  ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof

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

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

The TRS R consists of the following rules:

a(a(x1)) → b(a(b(x1)))
a(a(a(x1))) → a(b(a(b(a(x1)))))
a(b(a(x1))) → b(b(a(b(b(x1)))))
a(a(a(a(x1)))) → a(a(b(a(b(a(a(x1)))))))
a(a(b(a(x1)))) → a(b(b(a(b(a(b(x1)))))))
a(b(a(a(x1)))) → b(a(b(a(b(b(a(x1)))))))
a(b(b(a(x1)))) → b(b(b(a(b(b(b(x1)))))))
a(a(a(a(a(x1))))) → a(a(a(b(a(b(a(a(a(x1)))))))))
a(a(a(b(a(x1))))) → a(a(b(b(a(b(a(a(b(x1)))))))))
a(a(b(a(a(x1))))) → a(b(a(b(a(b(a(b(a(x1)))))))))
a(a(b(b(a(x1))))) → a(b(b(b(a(b(a(b(b(x1)))))))))
a(b(a(a(a(x1))))) → b(a(a(b(a(b(b(a(a(x1)))))))))
a(b(a(b(a(x1))))) → b(a(b(b(a(b(b(a(b(x1)))))))))
a(b(b(a(a(x1))))) → b(b(a(b(a(b(b(b(a(x1)))))))))
a(b(b(b(a(x1))))) → b(b(b(b(a(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(a(x1))))) → A(a(b(a(b(a(a(a(x1))))))))
A(a(a(x1))) → A(b(a(b(a(x1)))))
A(a(a(a(a(x1))))) → A(b(a(a(a(x1)))))
A(a(a(a(a(x1))))) → A(b(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(x1)))) → A(b(a(b(a(a(x1))))))
A(a(a(b(a(x1))))) → A(b(x1))
A(a(a(b(a(x1))))) → A(a(b(b(a(b(a(a(b(x1)))))))))
A(a(a(a(a(x1))))) → A(a(a(b(a(b(a(a(a(x1)))))))))
A(a(a(a(x1)))) → A(a(b(a(b(a(a(x1)))))))
A(a(a(b(a(x1))))) → A(b(b(a(b(a(a(b(x1))))))))
A(a(a(b(a(x1))))) → A(b(a(a(b(x1)))))
The remaining pairs can at least be oriented weakly.

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



↳ QTRS
  ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ QDP
                      ↳ Narrowing

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

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

The TRS R consists of the following rules:

a(a(x1)) → b(a(b(x1)))
a(a(a(x1))) → a(b(a(b(a(x1)))))
a(b(a(x1))) → b(b(a(b(b(x1)))))
a(a(a(a(x1)))) → a(a(b(a(b(a(a(x1)))))))
a(a(b(a(x1)))) → a(b(b(a(b(a(b(x1)))))))
a(b(a(a(x1)))) → b(a(b(a(b(b(a(x1)))))))
a(b(b(a(x1)))) → b(b(b(a(b(b(b(x1)))))))
a(a(a(a(a(x1))))) → a(a(a(b(a(b(a(a(a(x1)))))))))
a(a(a(b(a(x1))))) → a(a(b(b(a(b(a(a(b(x1)))))))))
a(a(b(a(a(x1))))) → a(b(a(b(a(b(a(b(a(x1)))))))))
a(a(b(b(a(x1))))) → a(b(b(b(a(b(a(b(b(x1)))))))))
a(b(a(a(a(x1))))) → b(a(a(b(a(b(b(a(a(x1)))))))))
a(b(a(b(a(x1))))) → b(a(b(b(a(b(b(a(b(x1)))))))))
a(b(b(a(a(x1))))) → b(b(a(b(a(b(b(b(a(x1)))))))))
a(b(b(b(a(x1))))) → b(b(b(b(a(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(b(a(x1))))) → A(b(b(a(b(b(a(b(x1)))))))) at position [0,0,0] we obtained the following new rules:

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



↳ QTRS
  ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof

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

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

The TRS R consists of the following rules:

a(a(x1)) → b(a(b(x1)))
a(a(a(x1))) → a(b(a(b(a(x1)))))
a(b(a(x1))) → b(b(a(b(b(x1)))))
a(a(a(a(x1)))) → a(a(b(a(b(a(a(x1)))))))
a(a(b(a(x1)))) → a(b(b(a(b(a(b(x1)))))))
a(b(a(a(x1)))) → b(a(b(a(b(b(a(x1)))))))
a(b(b(a(x1)))) → b(b(b(a(b(b(b(x1)))))))
a(a(a(a(a(x1))))) → a(a(a(b(a(b(a(a(a(x1)))))))))
a(a(a(b(a(x1))))) → a(a(b(b(a(b(a(a(b(x1)))))))))
a(a(b(a(a(x1))))) → a(b(a(b(a(b(a(b(a(x1)))))))))
a(a(b(b(a(x1))))) → a(b(b(b(a(b(a(b(b(x1)))))))))
a(b(a(a(a(x1))))) → b(a(a(b(a(b(b(a(a(x1)))))))))
a(b(a(b(a(x1))))) → b(a(b(b(a(b(b(a(b(x1)))))))))
a(b(b(a(a(x1))))) → b(b(a(b(a(b(b(b(a(x1)))))))))
a(b(b(b(a(x1))))) → b(b(b(b(a(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 5 less nodes.

↳ QTRS
  ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(a(b(a(x1)))) → A(b(b(a(b(a(b(x1))))))) at position [0,0,0] we obtained the following new rules:

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



↳ QTRS
  ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof

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

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

The TRS R consists of the following rules:

a(a(x1)) → b(a(b(x1)))
a(a(a(x1))) → a(b(a(b(a(x1)))))
a(b(a(x1))) → b(b(a(b(b(x1)))))
a(a(a(a(x1)))) → a(a(b(a(b(a(a(x1)))))))
a(a(b(a(x1)))) → a(b(b(a(b(a(b(x1)))))))
a(b(a(a(x1)))) → b(a(b(a(b(b(a(x1)))))))
a(b(b(a(x1)))) → b(b(b(a(b(b(b(x1)))))))
a(a(a(a(a(x1))))) → a(a(a(b(a(b(a(a(a(x1)))))))))
a(a(a(b(a(x1))))) → a(a(b(b(a(b(a(a(b(x1)))))))))
a(a(b(a(a(x1))))) → a(b(a(b(a(b(a(b(a(x1)))))))))
a(a(b(b(a(x1))))) → a(b(b(b(a(b(a(b(b(x1)))))))))
a(b(a(a(a(x1))))) → b(a(a(b(a(b(b(a(a(x1)))))))))
a(b(a(b(a(x1))))) → b(a(b(b(a(b(b(a(b(x1)))))))))
a(b(b(a(a(x1))))) → b(b(a(b(a(b(b(b(a(x1)))))))))
a(b(b(b(a(x1))))) → b(b(b(b(a(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 4 less nodes.

↳ QTRS
  ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ 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(a(b(a(a(x1))))) → A(b(a(b(a(b(a(b(a(x1)))))))))
A(a(b(a(x1)))) → A(b(x1))
A(b(a(a(a(x1))))) → A(b(a(b(b(a(a(x1)))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(b(a(b(a(a(a(a(x0)))))))) → A(b(b(a(b(b(b(a(a(b(a(b(b(a(a(x0)))))))))))))))
A(b(a(a(a(x1))))) → A(b(b(a(a(x1)))))
A(a(x1)) → A(b(x1))
A(a(b(a(a(a(a(x0))))))) → A(b(b(a(b(b(a(a(b(a(b(b(a(a(x0))))))))))))))
A(a(b(a(a(x0))))) → A(b(b(a(b(b(b(a(b(b(x0))))))))))
A(b(a(a(x1)))) → A(b(a(b(b(a(x1))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(a(b(a(a(b(a(x0))))))) → A(b(b(a(b(b(a(b(b(a(b(b(a(b(x0))))))))))))))
A(b(a(b(a(a(a(x0))))))) → A(b(b(a(b(b(b(a(b(a(b(b(a(x0)))))))))))))
A(b(a(a(a(x1))))) → A(a(b(a(b(b(a(a(x1))))))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(x1))) → A(b(b(x1)))
A(a(b(a(a(a(x0)))))) → A(b(b(a(b(b(a(b(a(b(b(a(x0))))))))))))
A(a(b(b(a(x1))))) → A(b(b(x1)))
A(b(a(b(a(x1))))) → A(b(x1))
A(b(b(a(a(x1))))) → A(b(a(b(b(b(a(x1)))))))
A(a(b(a(b(a(a(x0))))))) → A(b(b(a(b(b(b(a(b(a(b(b(b(a(x0))))))))))))))
A(a(b(b(a(x1))))) → A(b(a(b(b(x1)))))
A(b(a(b(a(a(b(a(x0)))))))) → A(b(b(a(b(b(b(a(b(b(a(b(b(a(b(x0)))))))))))))))
A(b(a(a(x1)))) → A(b(b(a(x1))))
A(a(b(a(x1)))) → A(b(a(b(x1))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(a(b(a(a(x0))))) → A(b(b(a(b(b(b(a(b(b(x0)))))))))) at position [0,0,0] we obtained the following new rules:

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



↳ QTRS
  ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ 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(a(b(a(a(x1))))) → A(b(a(b(a(b(a(b(a(x1)))))))))
A(a(b(a(x1)))) → A(b(x1))
A(b(a(a(a(x1))))) → A(b(a(b(b(a(a(x1)))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(a(b(a(a(b(a(x0))))))) → A(b(b(a(b(b(b(b(b(b(b(a(b(b(b(b(x0))))))))))))))))
A(b(a(b(a(a(a(a(x0)))))))) → A(b(b(a(b(b(b(a(a(b(a(b(b(a(a(x0)))))))))))))))
A(b(a(a(a(x1))))) → A(b(b(a(a(x1)))))
A(a(b(a(a(a(a(x0))))))) → A(b(b(a(b(b(a(a(b(a(b(b(a(a(x0))))))))))))))
A(a(x1)) → A(b(x1))
A(b(a(a(x1)))) → A(b(a(b(b(a(x1))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(x1)))))))
A(a(b(a(a(b(a(x0))))))) → A(b(b(a(b(b(a(b(b(a(b(b(a(b(x0))))))))))))))
A(b(a(b(a(a(a(x0))))))) → A(b(b(a(b(b(b(a(b(a(b(b(a(x0)))))))))))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(x1))))) → A(a(b(a(b(b(a(a(x1))))))))
A(b(a(x1))) → A(b(b(x1)))
A(a(b(a(a(a(x0)))))) → A(b(b(a(b(b(a(b(a(b(b(a(x0))))))))))))
A(a(b(a(a(y0))))) → A(b(b(b(b(b(b(a(b(b(b(b(b(b(y0))))))))))))))
A(a(b(a(a(a(x0)))))) → A(b(b(a(b(b(b(b(b(b(a(b(b(b(x0))))))))))))))
A(a(b(b(a(x1))))) → A(b(b(x1)))
A(b(a(b(a(x1))))) → A(b(x1))
A(a(b(a(b(a(a(x0))))))) → A(b(b(a(b(b(b(a(b(a(b(b(b(a(x0))))))))))))))
A(b(b(a(a(x1))))) → A(b(a(b(b(b(a(x1)))))))
A(a(b(b(a(x1))))) → A(b(a(b(b(x1)))))
A(a(b(a(a(a(a(x0))))))) → A(b(b(a(b(b(b(b(b(a(b(a(b(b(b(a(x0))))))))))))))))
A(b(a(b(a(a(b(a(x0)))))))) → A(b(b(a(b(b(b(a(b(b(a(b(b(a(b(x0)))))))))))))))
A(b(a(a(x1)))) → A(b(b(a(x1))))
A(a(b(a(x1)))) → A(b(a(b(x1))))

The TRS R consists of the following rules:

a(a(x1)) → b(a(b(x1)))
a(a(a(x1))) → a(b(a(b(a(x1)))))
a(b(a(x1))) → b(b(a(b(b(x1)))))
a(a(a(a(x1)))) → a(a(b(a(b(a(a(x1)))))))
a(a(b(a(x1)))) → a(b(b(a(b(a(b(x1)))))))
a(b(a(a(x1)))) → b(a(b(a(b(b(a(x1)))))))
a(b(b(a(x1)))) → b(b(b(a(b(b(b(x1)))))))
a(a(a(a(a(x1))))) → a(a(a(b(a(b(a(a(a(x1)))))))))
a(a(a(b(a(x1))))) → a(a(b(b(a(b(a(a(b(x1)))))))))
a(a(b(a(a(x1))))) → a(b(a(b(a(b(a(b(a(x1)))))))))
a(a(b(b(a(x1))))) → a(b(b(b(a(b(a(b(b(x1)))))))))
a(b(a(a(a(x1))))) → b(a(a(b(a(b(b(a(a(x1)))))))))
a(b(a(b(a(x1))))) → b(a(b(b(a(b(b(a(b(x1)))))))))
a(b(b(a(a(x1))))) → b(b(a(b(a(b(b(b(a(x1)))))))))
a(b(b(b(a(x1))))) → b(b(b(b(a(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 4 less nodes.

↳ QTRS
  ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ 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(a(b(a(a(b(a(x0))))))) → A(b(b(a(b(b(a(b(b(a(b(b(a(b(x0))))))))))))))
A(b(a(b(a(a(a(x0))))))) → A(b(b(a(b(b(b(a(b(a(b(b(a(x0)))))))))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(b(a(b(a(x1)))))))))
A(a(b(a(a(x1))))) → A(b(a(x1)))
A(b(a(a(a(x1))))) → A(a(b(a(b(b(a(a(x1))))))))
A(b(a(x1))) → A(b(b(x1)))
A(a(b(a(x1)))) → A(b(x1))
A(a(b(a(a(a(x0)))))) → A(b(b(a(b(b(a(b(a(b(b(a(x0))))))))))))
A(b(a(a(a(x1))))) → A(b(a(b(b(a(a(x1)))))))
A(a(b(a(a(x1))))) → A(b(a(b(a(x1)))))
A(b(a(b(a(a(a(a(x0)))))))) → A(b(b(a(b(b(b(a(a(b(a(b(b(a(a(x0)))))))))))))))
A(b(a(a(a(x1))))) → A(b(b(a(a(x1)))))
A(a(b(a(a(a(a(x0))))))) → A(b(b(a(b(b(a(a(b(a(b(b(a(a(x0))))))))))))))
A(a(b(b(a(x1))))) → A(b(b(x1)))
A(a(x1)) → A(b(x1))
A(b(a(b(a(x1))))) → A(b(x1))
A(b(a(a(x1)))) → A(b(a(b(b(a(x1))))))
A(a(b(a(b(a(a(x0))))))) → A(b(b(a(b(b(b(a(b(a(b(b(b(a(x0))))))))))))))
A(b(b(a(a(x1))))) → A(b(a(b(b(b(a(x1)))))))
A(a(b(b(a(x1))))) → A(b(a(b(b(x1)))))
A(b(a(b(a(a(b(a(x0)))))))) → A(b(b(a(b(b(b(a(b(b(a(b(b(a(b(x0)))))))))))))))
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(a(b(x1))))

The TRS R consists of the following rules:

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

The TRS R consists of the following rules:

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

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

↳ QTRS
  ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ 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.0(b.1(a.0(x1))))) → A.0(b.0(x1))
A.0(b.1(a.1(a.1(x1)))) → A.0(b.0(b.1(a.1(x1))))
A.1(a.0(b.0(b.1(a.0(x1))))) → A.0(b.0(b.0(x1)))
A.0(b.1(a.1(a.1(a.0(x1))))) → A.0(b.0(b.1(a.1(a.0(x1)))))
A.0(b.1(a.0(b.1(a.1(x1))))) → A.0(b.1(x1))
A.1(a.0(b.1(a.1(a.1(a.1(x0)))))) → A.0(b.0(b.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(x1))))) → A.0(b.1(a.0(b.0(b.1(a.1(a.1(x1)))))))
A.1(a.0(x1)) → A.0(b.0(x1))
A.1(a.0(b.1(a.0(b.1(a.1(a.1(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.1(x0))))))))))))))
A.0(b.1(a.1(a.1(a.0(x1))))) → A.1(a.0(b.1(a.0(b.0(b.1(a.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(b.1(a.0(b.1(a.0(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)))))))
A.0(b.1(a.1(a.1(a.1(x1))))) → A.0(b.0(b.1(a.1(a.1(x1)))))
A.0(b.1(a.0(b.1(a.1(a.0(b.1(a.1(x0)))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.1(x0)))))))))))))))
A.1(a.0(b.1(a.1(a.1(a.1(a.1(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.0(b.1(a.1(a.1(x0))))))))))))))
A.0(b.1(a.1(a.0(x1)))) → A.0(b.1(a.0(b.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.0(b.1(a.0(x1))))) → A.0(b.1(a.0(b.0(b.0(x1)))))
A.1(a.0(b.0(b.1(a.1(x1))))) → A.0(b.0(b.1(x1)))
A.0(b.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(b.0(b.0(b.1(a.0(x1)))))))
A.1(a.0(b.0(b.1(a.1(x1))))) → A.0(b.1(a.0(b.0(b.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.0(b.1(a.0(b.1(a.1(a.1(a.1(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(x0)))))))))))))
A.1(a.0(b.1(a.0(b.1(a.1(a.0(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(x0))))))))))))))
A.0(b.1(a.1(a.1(a.1(x1))))) → A.0(a.0(b.1(a.0(b.0(b.1(a.1(a.1(x1))))))))
A.1(a.1(x1)) → A.0(b.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.0(b.1(a.0(x1))) → A.0(b.0(b.0(x1)))
A.1(a.0(b.1(a.1(a.1(a.1(a.0(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(x0))))))))))))))
A.0(b.1(a.0(b.1(a.1(a.1(a.1(a.0(x0)))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(x0)))))))))))))))
A.0(b.1(a.1(a.1(a.0(x1))))) → A.0(b.1(a.0(b.0(b.1(a.1(a.0(x1)))))))
A.1(a.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(x1)))
A.0(b.1(a.0(b.1(a.1(a.1(a.0(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(x0)))))))))))))
A.1(a.0(b.1(a.1(a.0(b.1(a.1(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.1(x0))))))))))))))
A.1(a.0(b.1(a.1(x1)))) → A.0(b.1(x1))
A.0(b.1(a.1(a.0(x1)))) → A.0(b.0(b.1(a.0(x1))))
A.0(b.1(a.1(a.1(a.0(x1))))) → A.0(a.0(b.1(a.0(b.0(b.1(a.1(a.0(x1))))))))
A.0(b.1(a.1(a.1(a.1(x1))))) → A.1(a.0(b.1(a.0(b.0(b.1(a.1(a.1(x1))))))))
A.1(a.0(b.1(a.1(a.0(b.1(a.0(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(x0))))))))))))))
A.1(a.0(b.1(a.1(x1)))) → A.0(b.1(a.0(b.1(x1))))
A.1(a.0(b.1(a.1(a.1(a.0(x0)))))) → A.0(b.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.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.0(b.1(a.1(a.0(b.1(a.0(x0)))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(x0)))))))))))))))
A.0(b.1(a.1(x1))) → 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.0(b.1(a.1(x1)))))))))
A.0(b.1(a.0(b.1(a.1(a.1(a.1(a.1(x0)))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.1(a.0(b.1(a.0(b.0(b.1(a.1(a.1(x0)))))))))))))))
A.1(a.0(b.1(a.0(x1)))) → A.0(b.0(x1))
A.1(a.0(b.1(a.0(x1)))) → A.0(b.1(a.0(b.0(x1))))
A.0(b.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.0(b.0(b.0(b.1(a.1(x1)))))))
A.0(b.1(a.1(a.1(x1)))) → A.0(b.1(a.0(b.0(b.1(a.1(x1))))))

The TRS R consists of the following rules:

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

↳ QTRS
  ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ 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.0(b.1(a.0(x1))))) → A.0(b.0(x1))
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(x1)))) → A.0(b.0(b.1(a.1(x1))))
A.0(b.1(a.0(b.1(a.1(a.1(a.0(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(x0)))))))))))))
A.1(a.0(b.1(a.1(a.0(b.1(a.1(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.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.0(x1))))) → A.0(b.0(b.1(a.1(a.0(x1)))))
A.1(a.0(b.1(a.1(x1)))) → A.0(b.1(x1))
A.0(b.1(a.0(b.1(a.1(x1))))) → A.0(b.1(x1))
A.1(a.0(b.1(a.1(a.1(a.1(x0)))))) → A.0(b.0(b.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.0(x1))))) → A.0(a.0(b.1(a.0(b.0(b.1(a.1(a.0(x1))))))))
A.0(b.1(a.1(a.0(x1)))) → A.0(b.0(b.1(a.0(x1))))
A.0(b.1(a.1(a.1(a.1(x1))))) → A.0(b.1(a.0(b.0(b.1(a.1(a.1(x1)))))))
A.1(a.0(b.1(a.0(b.1(a.1(a.1(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.1(x0))))))))))))))
A.1(a.0(x1)) → A.0(b.0(x1))
A.0(b.1(a.1(a.1(a.0(x1))))) → A.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(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)))))))
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(b.1(a.0(x1)))))))))
A.0(b.1(a.1(a.1(a.1(x1))))) → A.1(a.0(b.1(a.0(b.0(b.1(a.1(a.1(x1))))))))
A.0(b.1(a.1(a.1(a.1(x1))))) → A.0(b.0(b.1(a.1(a.1(x1)))))
A.0(b.1(a.0(b.1(a.1(a.0(b.1(a.1(x0)))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.1(x0)))))))))))))))
A.1(a.0(b.1(a.1(a.1(a.1(a.1(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.0(b.1(a.1(a.1(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.0(x1)))) → A.0(b.1(a.0(b.0(b.1(a.0(x1))))))
A.1(a.0(b.0(b.1(a.1(x1))))) → A.0(b.0(b.1(x1)))
A.1(a.0(b.0(b.1(a.0(x1))))) → A.0(b.1(a.0(b.0(b.0(x1)))))
A.1(a.0(b.1(a.1(a.0(b.1(a.0(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(x0))))))))))))))
A.1(a.0(b.1(a.1(x1)))) → A.0(b.1(a.0(b.1(x1))))
A.0(b.0(b.1(a.1(a.0(x1))))) → A.0(b.1(a.0(b.0(b.0(b.1(a.0(x1)))))))
A.1(a.0(b.1(a.1(a.1(a.0(x0)))))) → A.0(b.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(x0))))))))))))
A.1(a.0(b.0(b.1(a.1(x1))))) → A.0(b.1(a.0(b.0(b.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.0(b.1(a.0(b.1(a.1(a.1(a.1(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(x0)))))))))))))
A.0(b.1(a.0(b.1(a.1(a.0(b.1(a.0(x0)))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(x0)))))))))))))))
A.1(a.0(b.1(a.0(b.1(a.1(a.0(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(a.0(b.0(b.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.0(b.1(a.1(x1)))))))))
A.0(b.1(a.1(x1))) → A.0(b.0(b.1(x1)))
A.0(b.1(a.1(a.1(a.1(x1))))) → A.0(a.0(b.1(a.0(b.0(b.1(a.1(a.1(x1))))))))
A.0(b.1(a.0(b.1(a.1(a.1(a.1(a.1(x0)))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.1(a.0(b.1(a.0(b.0(b.1(a.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.1(a.1(x1)) → A.0(b.1(x1))
A.1(a.0(b.1(a.0(x1)))) → A.0(b.0(x1))
A.1(a.0(b.1(a.0(x1)))) → A.0(b.1(a.0(b.0(x1))))
A.1(a.0(b.1(a.1(a.1(a.1(a.0(x0))))))) → A.0(b.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(x0))))))))))))))
A.0(b.1(a.0(b.1(a.1(a.1(a.1(a.0(x0)))))))) → A.0(b.0(b.1(a.0(b.0(b.0(b.1(a.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(x0)))))))))))))))
A.0(b.1(a.1(a.1(a.0(x1))))) → A.0(b.1(a.0(b.0(b.1(a.1(a.0(x1)))))))
A.0(b.0(b.1(a.1(a.1(x1))))) → A.0(b.1(a.0(b.0(b.0(b.1(a.1(x1)))))))
A.0(b.1(a.1(a.1(x1)))) → A.0(b.1(a.0(b.0(b.1(a.1(x1))))))

The TRS R consists of the following rules:

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

Strictly oriented rules of the TRS R:

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

Used ordering: POLO with Polynomial interpretation [25]:

POL(A.0(x1)) = 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
          ↳ Narrowing
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ 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.0(b.1(a.0(b.1(a.0(x1))))) → A.0(b.0(x1))
A.1(a.0(b.1(a.0(x1)))) → A.0(b.0(x1))
A.1(a.0(b.1(a.0(x1)))) → A.0(b.1(a.0(b.0(x1))))
A.1(a.0(b.0(b.1(a.0(x1))))) → A.0(b.1(a.0(b.0(b.0(x1)))))
A.1(a.0(x1)) → A.0(b.0(x1))

The TRS R consists of the following rules:

a.1(a.1(a.1(a.1(a.0(x1))))) → a.1(a.1(a.0(b.1(a.0(b.1(a.1(a.1(a.0(x1)))))))))
a.0(b.1(a.1(a.1(a.1(x1))))) → b.1(a.1(a.0(b.1(a.0(b.0(b.1(a.1(a.1(x1)))))))))
b.1(x0) → b.0(x0)
a.0(b.0(b.0(b.1(a.0(x1))))) → b.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.0(x1)))))))))
a.1(a.1(a.1(a.1(a.1(x1))))) → a.1(a.1(a.0(b.1(a.0(b.1(a.1(a.1(a.1(x1)))))))))
a.1(a.1(a.0(b.1(a.0(x1))))) → a.1(a.0(b.0(b.1(a.0(b.1(a.1(a.0(b.0(x1)))))))))
a.0(b.1(a.0(x1))) → b.0(b.1(a.0(b.0(b.0(x1)))))
a.0(b.1(a.0(b.1(a.0(x1))))) → b.1(a.0(b.0(b.1(a.0(b.0(b.1(a.0(b.0(x1)))))))))
a.0(b.0(b.1(a.0(x1)))) → b.0(b.0(b.1(a.0(b.0(b.0(b.0(x1)))))))
a.0(b.1(a.1(a.1(a.0(x1))))) → b.1(a.1(a.0(b.1(a.0(b.0(b.1(a.1(a.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 5 less nodes.