Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

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

Q is empty.


QTRS
  ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

Q is empty.

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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

B(a(a(b(a(a(b(x0))))))) → B(a(a(a(b(a(a(a(b(x0)))))))))
B(a(a(b(a(a(b(x0))))))) → B(a(a(a(x0))))
B(a(a(b(y0)))) → B(b(b(b(b(y0)))))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
QDP
                  ↳ QDPToSRSProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The finiteness of this DP problem is implied by strong termination of a SRS due to [12].


↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ QDPToSRSProof
QTRS
                      ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

Q is empty.

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

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

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

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

The set Q is empty.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ QDPToSRSProof
                    ↳ QTRS
                      ↳ QTRS Reverse
QTRS
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
                          ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.

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

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
A(a(a(x))) → B(x)
A(a(a(a(a(b(x)))))) → B(b(a(a(a(b(x))))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(b(a(a(a(b(x)))))))))
B(a(a(b(x)))) → B(b(b(b(b(x)))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(x))))
A(a(a(a(a(b(x)))))) → B(b(b(a(a(a(b(x)))))))
A(a(a(a(a(b(x)))))) → B(x)
B(a(a(b(x)))) → A(a(a(b(x))))
A(a(a(a(a(b(x)))))) → B(b(x))

The set Q is empty.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ QDPToSRSProof
                    ↳ QTRS
                      ↳ QTRS Reverse
                        ↳ QTRS
                          ↳ QTRS Reverse
QTRS
                          ↳ QTRS Reverse
                          ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
A(a(a(x))) → B(x)
A(a(a(a(a(b(x)))))) → B(b(a(a(a(b(x))))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(b(a(a(a(b(x)))))))))
B(a(a(b(x)))) → B(b(b(b(b(x)))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(x))))
A(a(a(a(a(b(x)))))) → B(b(b(a(a(a(b(x)))))))
A(a(a(a(a(b(x)))))) → B(x)
B(a(a(b(x)))) → A(a(a(b(x))))
A(a(a(a(a(b(x)))))) → B(b(x))

Q is empty.

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

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
A(a(a(x))) → B(x)
A(a(a(a(a(b(x)))))) → B(b(a(a(a(b(x))))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(b(a(a(a(b(x)))))))))
B(a(a(b(x)))) → B(b(b(b(b(x)))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(x))))
A(a(a(a(a(b(x)))))) → B(b(b(a(a(a(b(x)))))))
A(a(a(a(a(b(x)))))) → B(x)
B(a(a(b(x)))) → A(a(a(b(x))))
A(a(a(a(a(b(x)))))) → B(b(x))

The set Q is empty.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ QDPToSRSProof
                    ↳ QTRS
                      ↳ QTRS Reverse
                        ↳ QTRS
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
QTRS
                          ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
A(a(a(x))) → B(x)
A(a(a(a(a(b(x)))))) → B(b(a(a(a(b(x))))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(b(a(a(a(b(x)))))))))
B(a(a(b(x)))) → B(b(b(b(b(x)))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(x))))
A(a(a(a(a(b(x)))))) → B(b(b(a(a(a(b(x)))))))
A(a(a(a(a(b(x)))))) → B(x)
B(a(a(b(x)))) → A(a(a(b(x))))
A(a(a(a(a(b(x)))))) → 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:

B1(a(a(B(x)))) → B1(b(b(b(B(x)))))
B1(a(a(a(a(A(x)))))) → B1(B(x))
B1(a(a(a(a(A(x)))))) → A1(a(b(B(x))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(B(x)))) → B1(B(x))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(a(a(A(x)))))) → A1(b(b(B(x))))
B1(a(a(B(x)))) → A1(a(A(x)))
B1(a(a(a(a(A(x)))))) → A1(b(B(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(b(b(B(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(b(a(a(a(B(x)))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(B(x)))) → A1(A(x))
B1(a(a(B(x)))) → B1(b(b(B(x))))
B1(a(a(B(x)))) → B1(b(B(x)))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(b(B(x)))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

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

B1(a(a(B(x)))) → B1(b(b(b(B(x)))))
B1(a(a(a(a(A(x)))))) → B1(B(x))
B1(a(a(a(a(A(x)))))) → A1(a(b(B(x))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(B(x)))) → B1(B(x))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(a(a(A(x)))))) → A1(b(b(B(x))))
B1(a(a(B(x)))) → A1(a(A(x)))
B1(a(a(a(a(A(x)))))) → A1(b(B(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(b(b(B(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(b(a(a(a(B(x)))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(B(x)))) → A1(A(x))
B1(a(a(B(x)))) → B1(b(b(B(x))))
B1(a(a(B(x)))) → B1(b(B(x)))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(b(B(x)))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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 12 less nodes.

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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(b(a(a(a(B(x)))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(B(x)))))) at position [0] we obtained the following new rules:

B1(a(a(a(a(A(y0)))))) → B1(b(b(b(b(B(y0))))))



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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(b(a(a(a(B(x)))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(a(a(A(y0)))))) → B1(b(b(b(b(B(y0))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(b(a(a(a(B(x)))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(B(x))))))) → A1(a(b(a(a(a(B(x))))))) at position [0] we obtained the following new rules:

B1(a(a(b(a(a(B(y0))))))) → A1(a(b(b(b(b(B(y0)))))))



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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(a(a(B(y0))))))) → A1(a(b(b(b(b(B(y0)))))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x)))))) at position [0] we obtained the following new rules:

B1(a(a(b(a(a(B(y0))))))) → A1(b(b(b(b(B(y0))))))



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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(y0))))))) → A1(b(b(b(b(B(y0))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x))))) at position [0] we obtained the following new rules:

B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(B(y0)))))



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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(B(y0)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x))))))) at position [0] we obtained the following new rules:

B1(a(a(a(a(A(y0)))))) → B1(b(b(b(b(b(B(y0)))))))



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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(a(a(A(y0)))))) → B1(b(b(b(b(b(B(y0)))))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
A1(a(a(x))) → B1(b(x))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(B(x)))) → B1(a(a(A(x)))) at position [0] we obtained the following new rules:

B1(a(a(B(x0)))) → B1(B(x0))



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

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

B1(a(a(B(x0)))) → B1(B(x0))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
A1(a(a(x))) → B1(b(x))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
A1(a(a(x))) → B1(b(x))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(x)))) → B1(a(a(a(b(x))))) at position [0] we obtained the following new rules:

B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))



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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(x))) → B1(b(b(x))) at position [0] we obtained the following new rules:

A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(B(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(B(x0)))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(B(x0))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(B(x0)))))) → B1(b(b(b(b(b(B(x0)))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))



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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(b(b(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(B(x0)))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(B(x0))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(B(x0))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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 3 less nodes.

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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(B(x0))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(x))) → B1(b(x)) at position [0] we obtained the following new rules:

A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(B(x0)))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(b(b(B(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(B(x0))



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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(B(x0)))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(B(x0)))))) → B1(b(b(b(b(B(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(B(x0))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(B(x0))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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 3 less nodes.

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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(B(x0))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(B(x0)))))) at position [0] we obtained the following new rules:

A1(a(a(a(a(b(a(a(B(y0))))))))) → B1(b(b(b(b(B(y0))))))



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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
A1(a(a(a(a(b(a(a(B(y0))))))))) → B1(b(b(b(b(B(y0))))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x))))))))) at position [0] we obtained the following new rules:

B1(a(a(b(a(a(B(y0))))))) → B1(a(a(a(b(b(b(b(B(y0)))))))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))



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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0)))))) at position [0] we obtained the following new rules:

A1(a(a(a(a(B(x0)))))) → B1(b(b(B(x0))))



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

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

A1(a(a(a(a(B(x0)))))) → B1(b(b(B(x0))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0)))))))) at position [0] we obtained the following new rules:

A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(b(B(y0))))))))



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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(b(B(y0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0))))) at position [0] we obtained the following new rules:

B1(a(a(b(a(a(a(a(A(y0))))))))) → B1(b(b(b(B(y0)))))



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

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

B1(a(a(b(a(a(a(a(A(y0))))))))) → B1(b(b(b(B(y0)))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0))))))))) at position [0] we obtained the following new rules:

A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(b(b(B(y0)))))))))



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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(b(b(B(y0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0)))))) at position [0] we obtained the following new rules:

B1(a(a(b(a(a(a(a(A(y0))))))))) → B1(b(b(b(b(B(y0))))))



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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(a(a(A(y0))))))))) → B1(b(b(b(b(B(y0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0))))))))) at position [0] we obtained the following new rules:

B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(b(b(b(B(y0)))))))))



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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(b(b(b(B(y0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0))))))))) at position [0] we obtained the following new rules:

B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(b(b(b(B(y0)))))))))



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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(b(b(b(B(y0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0))))) at position [0] we obtained the following new rules:

A1(a(a(a(a(B(x0)))))) → B1(b(B(x0)))



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

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

B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(B(x0)))))) → B1(b(B(x0)))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0))))) at position [0] we obtained the following new rules:

A1(a(a(a(a(b(a(a(B(y0))))))))) → B1(b(b(b(B(y0)))))



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

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

B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(a(a(B(y0))))))))) → B1(b(b(b(B(y0)))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0)))))))) at position [0] we obtained the following new rules:

A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(b(B(y0))))))))



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

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

B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(b(B(y0))))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0))))))) at position [0] we obtained the following new rules:

A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(B(y0)))))))



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

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

B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(B(y0)))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

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

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

B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)

The TRS R consists of the following rules:

a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We have reversed the following QTRS:
The set of rules R is

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

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

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

The set Q is empty.

↳ QTRS
  ↳ DependencyPairsProof
  ↳ QTRS Reverse
QTRS
  ↳ QTRS Reverse

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

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

Q is empty.

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

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

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

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

The set Q is empty.

↳ QTRS
  ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse
QTRS

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

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

Q is empty.