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

Q is empty.


QTRS
  ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

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

The TRS R consists of the following rules:

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

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

The TRS R consists of the following rules:

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

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

B(b(b(b(a(x0))))) → B(b(b(a(x0))))



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

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

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

The TRS R consists of the following rules:

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

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

B(b(b(a(x0)))) → A(a(b(b(a(x0)))))



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

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

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

The TRS R consists of the following rules:

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

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

B(b(b(a(x0)))) → A(b(b(a(x0))))



↳ 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:

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

The TRS R consists of the following rules:

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

Q is empty.

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

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

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

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

The set Q is empty.

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

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

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

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

The TRS R consists of the following rules:

a(b(a(x))) → a(b(b(x)))
b(b(b(x))) → a(b(x))
b(b(x)) → a(a(a(x)))
b(B(x)) → A(x)
a(b(b(b(B(x))))) → a(b(b(B(x))))
a(b(A(x))) → a(b(B(x)))
a(b(b(B(x)))) → a(b(b(a(A(x)))))
a(b(b(B(x)))) → a(b(b(A(x))))
b(b(B(x))) → A(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
                          ↳ DependencyPairsProof
QDP
                              ↳ DependencyGraphProof
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

a(b(a(x))) → a(b(b(x)))
b(b(b(x))) → a(b(x))
b(b(x)) → a(a(a(x)))
b(B(x)) → A(x)
a(b(b(b(B(x))))) → a(b(b(B(x))))
a(b(A(x))) → a(b(B(x)))
a(b(b(B(x)))) → a(b(b(a(A(x)))))
a(b(b(B(x)))) → a(b(b(A(x))))
b(b(B(x))) → A(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 6 less nodes.

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

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

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

The TRS R consists of the following rules:

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

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

A1(b(A(x0))) → A1(A(x0))



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

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

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

The TRS R consists of the following rules:

a(b(a(x))) → a(b(b(x)))
b(b(b(x))) → a(b(x))
b(b(x)) → a(a(a(x)))
b(B(x)) → A(x)
a(b(b(b(B(x))))) → a(b(b(B(x))))
a(b(A(x))) → a(b(B(x)))
a(b(b(B(x)))) → a(b(b(a(A(x)))))
a(b(b(B(x)))) → a(b(b(A(x))))
b(b(B(x))) → A(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
                          ↳ DependencyPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
QDP
                                          ↳ Narrowing
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

a(b(a(x))) → a(b(b(x)))
b(b(b(x))) → a(b(x))
b(b(x)) → a(a(a(x)))
b(B(x)) → A(x)
a(b(b(b(B(x))))) → a(b(b(B(x))))
a(b(A(x))) → a(b(B(x)))
a(b(b(B(x)))) → a(b(b(a(A(x)))))
a(b(b(B(x)))) → a(b(b(A(x))))
b(b(B(x))) → A(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
                          ↳ DependencyPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
QDP
                                                  ↳ Narrowing
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

A1(b(b(B(y0)))) → A1(a(a(a(A(y0)))))



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

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

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

The TRS R consists of the following rules:

a(b(a(x))) → a(b(b(x)))
b(b(b(x))) → a(b(x))
b(b(x)) → a(a(a(x)))
b(B(x)) → A(x)
a(b(b(b(B(x))))) → a(b(b(B(x))))
a(b(A(x))) → a(b(B(x)))
a(b(b(B(x)))) → a(b(b(a(A(x)))))
a(b(b(B(x)))) → a(b(b(A(x))))
b(b(B(x))) → A(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
                          ↳ DependencyPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
QDP
                                                          ↳ ForwardInstantiation
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule A1(b(a(x))) → B1(x) we obtained the following new rules:

A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(y_0)))) → B1(b(y_0))



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

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

A1(b(a(x))) → B1(b(x))
A1(b(b(b(B(x))))) → A1(b(b(B(x))))
A1(b(a(x))) → A1(b(b(x)))
A1(b(b(B(x)))) → B1(b(a(A(x))))
A1(b(b(B(x)))) → B1(b(A(x)))
B1(b(x)) → A1(x)
B1(b(b(x))) → A1(b(x))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule B1(b(x)) → A1(x) we obtained the following new rules:

B1(b(b(b(b(B(y_0)))))) → A1(b(b(b(B(y_0)))))
B1(b(b(y_1))) → A1(b(y_1))
B1(b(b(b(B(y_0))))) → A1(b(b(B(y_0))))



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

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

A1(b(a(x))) → B1(b(x))
A1(b(b(b(B(x))))) → A1(b(b(B(x))))
A1(b(a(x))) → A1(b(b(x)))
A1(b(b(B(x)))) → B1(b(A(x)))
A1(b(b(B(x)))) → B1(b(a(A(x))))
B1(b(b(b(b(B(y_0)))))) → A1(b(b(b(B(y_0)))))
B1(b(b(x))) → A1(b(x))
B1(b(b(b(B(y_0))))) → A1(b(b(B(y_0))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

a(b(a(x))) → a(b(b(x)))
b(b(b(x))) → a(b(x))
b(b(x)) → a(a(a(x)))
b(B(x)) → A(x)
a(b(b(b(B(x))))) → a(b(b(B(x))))
a(b(A(x))) → a(b(B(x)))
a(b(b(B(x)))) → a(b(b(a(A(x)))))
a(b(b(B(x)))) → a(b(b(A(x))))
b(b(B(x))) → A(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 2 less nodes.

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

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

A1(b(a(x))) → B1(b(x))
A1(b(b(b(B(x))))) → A1(b(b(B(x))))
A1(b(a(x))) → A1(b(b(x)))
B1(b(b(b(b(B(y_0)))))) → A1(b(b(b(B(y_0)))))
B1(b(b(x))) → A1(b(x))
B1(b(b(b(B(y_0))))) → A1(b(b(B(y_0))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

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

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

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



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

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

A1(b(a(x))) → B1(b(x))
A1(b(b(b(B(y0))))) → A1(a(a(a(B(y0)))))
A1(b(a(x))) → A1(b(b(x)))
A1(b(b(b(B(x0))))) → A1(b(A(x0)))
B1(b(b(b(b(B(y_0)))))) → A1(b(b(b(B(y_0)))))
B1(b(b(x))) → A1(b(x))
A1(b(b(b(B(x0))))) → A1(A(x0))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
B1(b(b(b(B(y_0))))) → A1(b(b(B(y_0))))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

a(b(a(x))) → a(b(b(x)))
b(b(b(x))) → a(b(x))
b(b(x)) → a(a(a(x)))
b(B(x)) → A(x)
a(b(b(b(B(x))))) → a(b(b(B(x))))
a(b(A(x))) → a(b(B(x)))
a(b(b(B(x)))) → a(b(b(a(A(x)))))
a(b(b(B(x)))) → a(b(b(A(x))))
b(b(B(x))) → A(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
                          ↳ DependencyPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ ForwardInstantiation
                                                            ↳ QDP
                                                              ↳ ForwardInstantiation
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
QDP
                                                                              ↳ Narrowing
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

A1(b(a(x))) → B1(b(x))
A1(b(a(x))) → A1(b(b(x)))
B1(b(b(b(b(B(y_0)))))) → A1(b(b(b(B(y_0)))))
B1(b(b(x))) → A1(b(x))
B1(b(b(b(B(y_0))))) → A1(b(b(B(y_0))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

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

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

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



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

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

A1(b(a(x))) → B1(b(x))
B1(b(b(b(b(B(y_0)))))) → A1(b(b(b(B(y_0)))))
A1(b(a(x0))) → A1(a(a(a(x0))))
A1(b(a(b(B(x0))))) → A1(b(A(x0)))
A1(b(a(B(x0)))) → A1(A(x0))
A1(b(a(b(x0)))) → A1(a(b(x0)))
A1(b(a(B(x0)))) → A1(b(A(x0)))
B1(b(b(x))) → A1(b(x))
A1(b(a(b(b(x0))))) → A1(b(a(b(x0))))
B1(b(b(b(B(y_0))))) → A1(b(b(B(y_0))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(x0)))) → A1(b(a(a(a(x0)))))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

a(b(a(x))) → a(b(b(x)))
b(b(b(x))) → a(b(x))
b(b(x)) → a(a(a(x)))
b(B(x)) → A(x)
a(b(b(b(B(x))))) → a(b(b(B(x))))
a(b(A(x))) → a(b(B(x)))
a(b(b(B(x)))) → a(b(b(a(A(x)))))
a(b(b(B(x)))) → a(b(b(A(x))))
b(b(B(x))) → A(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 5 less nodes.

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

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

A1(b(a(x))) → B1(b(x))
B1(b(b(b(b(B(y_0)))))) → A1(b(b(b(B(y_0)))))
A1(b(a(b(b(x0))))) → A1(b(a(b(x0))))
B1(b(b(x))) → A1(b(x))
A1(b(a(b(x0)))) → A1(b(a(a(a(x0)))))
B1(b(b(b(B(y_0))))) → A1(b(b(B(y_0))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(b(b(b(b(B(y_0)))))) → A1(b(b(b(B(y_0))))) at position [0] we obtained the following new rules:

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



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

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

B1(b(b(b(b(B(y0)))))) → A1(b(a(a(a(B(y0))))))
A1(b(a(x))) → B1(b(x))
B1(b(b(b(b(B(y0)))))) → A1(a(b(B(y0))))
B1(b(b(b(b(B(y0)))))) → A1(a(a(a(b(B(y0))))))
B1(b(b(b(b(B(x0)))))) → A1(b(b(A(x0))))
A1(b(a(b(b(x0))))) → A1(b(a(b(x0))))
B1(b(b(x))) → A1(b(x))
A1(b(a(b(x0)))) → A1(b(a(a(a(x0)))))
B1(b(b(b(B(y_0))))) → A1(b(b(B(y_0))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
B1(b(b(b(b(B(x0)))))) → A1(b(A(x0)))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

a(b(a(x))) → a(b(b(x)))
b(b(b(x))) → a(b(x))
b(b(x)) → a(a(a(x)))
b(B(x)) → A(x)
a(b(b(b(B(x))))) → a(b(b(B(x))))
a(b(A(x))) → a(b(B(x)))
a(b(b(B(x)))) → a(b(b(a(A(x)))))
a(b(b(B(x)))) → a(b(b(A(x))))
b(b(B(x))) → A(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
                          ↳ DependencyPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ ForwardInstantiation
                                                            ↳ QDP
                                                              ↳ ForwardInstantiation
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
QDP
                                                                                              ↳ Narrowing
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(b(b(b(b(B(y0)))))) → A1(b(a(a(a(B(y0))))))
A1(b(a(x))) → B1(b(x))
B1(b(b(b(b(B(x0)))))) → A1(b(b(A(x0))))
A1(b(a(b(b(x0))))) → A1(b(a(b(x0))))
B1(b(b(x))) → A1(b(x))
A1(b(a(b(x0)))) → A1(b(a(a(a(x0)))))
B1(b(b(b(B(y_0))))) → A1(b(b(B(y_0))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(b(b(b(B(y_0))))) → A1(b(b(B(y_0)))) at position [0] we obtained the following new rules:

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



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

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

B1(b(b(b(B(y0))))) → A1(a(a(a(B(y0)))))
B1(b(b(b(b(B(y0)))))) → A1(b(a(a(a(B(y0))))))
A1(b(a(x))) → B1(b(x))
B1(b(b(b(B(x0))))) → A1(A(x0))
B1(b(b(b(B(x0))))) → A1(b(A(x0)))
B1(b(b(b(b(B(x0)))))) → A1(b(b(A(x0))))
B1(b(b(x))) → A1(b(x))
A1(b(a(b(b(x0))))) → A1(b(a(b(x0))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(x0)))) → A1(b(a(a(a(x0)))))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

a(b(a(x))) → a(b(b(x)))
b(b(b(x))) → a(b(x))
b(b(x)) → a(a(a(x)))
b(B(x)) → A(x)
a(b(b(b(B(x))))) → a(b(b(B(x))))
a(b(A(x))) → a(b(B(x)))
a(b(b(B(x)))) → a(b(b(a(A(x)))))
a(b(b(B(x)))) → a(b(b(A(x))))
b(b(B(x))) → A(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
                          ↳ DependencyPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ ForwardInstantiation
                                                            ↳ QDP
                                                              ↳ ForwardInstantiation
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
QDP
                                                                                                      ↳ Narrowing
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(b(b(b(b(B(y0)))))) → A1(b(a(a(a(B(y0))))))
A1(b(a(x))) → B1(b(x))
B1(b(b(b(b(B(x0)))))) → A1(b(b(A(x0))))
A1(b(a(b(b(x0))))) → A1(b(a(b(x0))))
B1(b(b(x))) → A1(b(x))
A1(b(a(b(x0)))) → A1(b(a(a(a(x0)))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

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

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

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



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

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

B1(b(b(b(b(B(y0)))))) → A1(b(a(a(a(B(y0))))))
A1(b(a(x))) → B1(b(x))
B1(b(b(b(b(B(y0)))))) → A1(a(a(a(A(y0)))))
B1(b(b(x))) → A1(b(x))
A1(b(a(b(b(x0))))) → A1(b(a(b(x0))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(x0)))) → A1(b(a(a(a(x0)))))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

a(b(a(x))) → a(b(b(x)))
b(b(b(x))) → a(b(x))
b(b(x)) → a(a(a(x)))
b(B(x)) → A(x)
a(b(b(b(B(x))))) → a(b(b(B(x))))
a(b(A(x))) → a(b(B(x)))
a(b(b(B(x)))) → a(b(b(a(A(x)))))
a(b(b(B(x)))) → a(b(b(A(x))))
b(b(B(x))) → A(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
                          ↳ DependencyPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ ForwardInstantiation
                                                            ↳ QDP
                                                              ↳ ForwardInstantiation
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
QDP
                                                                                                              ↳ ForwardInstantiation
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(b(b(b(b(B(y0)))))) → A1(b(a(a(a(B(y0))))))
A1(b(a(x))) → B1(b(x))
A1(b(a(b(b(x0))))) → A1(b(a(b(x0))))
B1(b(b(x))) → A1(b(x))
A1(b(a(b(x0)))) → A1(b(a(a(a(x0)))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule A1(b(a(x))) → B1(b(x)) we obtained the following new rules:

A1(b(a(b(b(b(B(y_0))))))) → B1(b(b(b(b(B(y_0))))))
A1(b(a(b(y_0)))) → B1(b(b(y_0)))



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

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

B1(b(b(b(b(B(y0)))))) → A1(b(a(a(a(B(y0))))))
A1(b(a(b(b(b(B(y_0))))))) → B1(b(b(b(b(B(y_0))))))
A1(b(a(b(y_0)))) → B1(b(b(y_0)))
B1(b(b(x))) → A1(b(x))
A1(b(a(b(b(x0))))) → A1(b(a(b(x0))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(x0)))) → A1(b(a(a(a(x0)))))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

a(b(a(x))) → a(b(b(x)))
b(b(b(x))) → a(b(x))
b(b(x)) → a(a(a(x)))
b(B(x)) → A(x)
a(b(b(b(B(x))))) → a(b(b(B(x))))
a(b(A(x))) → a(b(B(x)))
a(b(b(B(x)))) → a(b(b(a(A(x)))))
a(b(b(B(x)))) → a(b(b(A(x))))
b(b(B(x))) → A(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
                          ↳ DependencyPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ ForwardInstantiation
                                                            ↳ QDP
                                                              ↳ ForwardInstantiation
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ ForwardInstantiation
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
QDP
                                                                                                                      ↳ ForwardInstantiation
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

A1(b(a(b(b(b(B(y_0))))))) → B1(b(b(b(b(B(y_0))))))
A1(b(a(b(y_0)))) → B1(b(b(y_0)))
A1(b(a(b(b(x0))))) → A1(b(a(b(x0))))
B1(b(b(x))) → A1(b(x))
A1(b(a(b(x0)))) → A1(b(a(a(a(x0)))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(y_0)))) → B1(b(y_0))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule A1(b(a(b(y_0)))) → B1(b(y_0)) we obtained the following new rules:

A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ QDPToSRSProof
                    ↳ QTRS
                      ↳ QTRS Reverse
                        ↳ QTRS
                          ↳ DependencyPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ ForwardInstantiation
                                                            ↳ QDP
                                                              ↳ ForwardInstantiation
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ ForwardInstantiation
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ ForwardInstantiation
QDP
                                                                                                                          ↳ SemLabProof
                                                                                                                          ↳ SemLabProof2
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

A1(b(a(b(b(b(B(y_0))))))) → B1(b(b(b(b(B(y_0))))))
A1(b(a(b(y_0)))) → B1(b(b(y_0)))
B1(b(b(x))) → A1(b(x))
A1(b(a(b(b(x0))))) → A1(b(a(b(x0))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))
A1(b(a(b(x0)))) → A1(b(a(a(a(x0)))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We found the following quasi-model for the rules of the TRS R. Interpretation over the domain with elements from 0 to 1.A1: 0
B: 0
a: 0
A: 0
B1: 0
b: 1
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:

A1.1(b.0(a.1(b.1(b.1(b.0(B.1(y_0))))))) → B1.0(b.1(b.1(b.1(b.0(B.1(y_0))))))
A1.1(b.0(a.1(b.1(x0)))) → A1.1(b.0(a.0(a.0(a.1(x0)))))
B1.1(b.1(b.0(x))) → A1.0(b.0(x))
A1.1(b.0(a.1(b.1(b.0(x0))))) → A1.1(b.0(a.1(b.0(x0))))
A1.1(b.0(a.1(b.0(y_0)))) → B1.0(b.1(b.0(y_0)))
A1.1(b.0(a.1(b.1(b.0(y_0))))) → B1.0(b.1(b.0(y_0)))
A1.1(b.0(a.1(b.1(b.1(b.0(B.0(y_0))))))) → B1.1(b.1(b.1(b.1(b.0(B.0(y_0))))))
A1.1(b.0(a.1(b.1(b.1(b.0(B.1(y_0))))))) → B1.1(b.1(b.1(b.1(b.0(B.1(y_0))))))
A1.1(b.0(a.1(b.1(b.1(y_0))))) → B1.0(b.1(b.1(y_0)))
A1.1(b.0(a.1(b.1(b.1(b.0(B.0(y_0))))))) → B1.0(b.1(b.1(b.1(b.0(B.0(y_0))))))
A1.1(b.0(a.1(b.1(x0)))) → A1.0(b.0(a.0(a.0(a.1(x0)))))
B1.1(b.1(b.0(x))) → A1.1(b.0(x))
B1.1(b.1(b.1(x))) → A1.1(b.1(x))
A1.1(b.0(a.1(b.1(b.1(x0))))) → A1.0(b.0(a.1(b.1(x0))))
A1.1(b.0(a.1(b.0(y_0)))) → B1.1(b.1(b.0(y_0)))
A1.1(b.0(a.1(b.1(y_0)))) → B1.1(b.1(b.1(y_0)))
A1.1(b.0(a.1(b.0(x0)))) → A1.0(b.0(a.0(a.0(a.0(x0)))))
A1.1(b.0(a.1(b.1(y_0)))) → B1.0(b.1(b.1(y_0)))
A1.1(b.0(a.1(b.1(b.1(x0))))) → A1.1(b.0(a.1(b.1(x0))))
B1.1(b.1(b.1(x))) → A1.0(b.1(x))
A1.1(b.0(a.1(b.0(x0)))) → A1.1(b.0(a.0(a.0(a.0(x0)))))
A1.1(b.0(a.1(b.1(b.0(y_0))))) → B1.1(b.1(b.0(y_0)))
A1.1(b.0(a.1(b.1(b.0(x0))))) → A1.0(b.0(a.1(b.0(x0))))
A1.1(b.0(a.1(b.1(b.1(y_0))))) → B1.1(b.1(b.1(y_0)))

The TRS R consists of the following rules:

A.1(x0) → A.0(x0)
b.1(b.0(x)) → a.0(a.0(a.0(x)))
a.1(b.1(b.0(B.0(x)))) → a.1(b.1(b.0(A.0(x))))
b.0(B.0(x)) → A.0(x)
a.1(b.0(A.0(x))) → a.1(b.0(B.0(x)))
b.1(x0) → b.0(x0)
a.1(b.1(b.0(B.0(x)))) → a.1(b.1(b.0(a.0(A.0(x)))))
b.1(b.1(x)) → a.0(a.0(a.1(x)))
a.1(b.1(b.1(b.0(B.1(x))))) → a.1(b.1(b.0(B.1(x))))
a.1(x0) → a.0(x0)
a.1(b.0(a.0(x))) → a.1(b.1(b.0(x)))
b.1(b.1(b.1(x))) → a.1(b.1(x))
b.0(B.1(x)) → A.1(x)
b.1(b.1(b.0(x))) → a.1(b.0(x))
a.1(b.0(a.1(x))) → a.1(b.1(b.1(x)))
B.1(x0) → B.0(x0)
b.1(b.0(B.0(x))) → A.0(x)
a.1(b.1(b.1(b.0(B.0(x))))) → a.1(b.1(b.0(B.0(x))))
a.1(b.1(b.0(B.1(x)))) → a.1(b.1(b.0(A.1(x))))
a.1(b.0(A.1(x))) → a.1(b.0(B.1(x)))
b.1(b.0(B.1(x))) → A.1(x)
a.1(b.1(b.0(B.1(x)))) → a.1(b.1(b.0(a.0(A.1(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
                          ↳ DependencyPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ ForwardInstantiation
                                                            ↳ QDP
                                                              ↳ ForwardInstantiation
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ ForwardInstantiation
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ ForwardInstantiation
                                                                                                                        ↳ QDP
                                                                                                                          ↳ SemLabProof
QDP
                                                                                                                              ↳ DependencyGraphProof
                                                                                                                          ↳ SemLabProof2
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

A1.1(b.0(a.1(b.1(b.1(b.0(B.1(y_0))))))) → B1.0(b.1(b.1(b.1(b.0(B.1(y_0))))))
A1.1(b.0(a.1(b.1(x0)))) → A1.1(b.0(a.0(a.0(a.1(x0)))))
B1.1(b.1(b.0(x))) → A1.0(b.0(x))
A1.1(b.0(a.1(b.1(b.0(x0))))) → A1.1(b.0(a.1(b.0(x0))))
A1.1(b.0(a.1(b.0(y_0)))) → B1.0(b.1(b.0(y_0)))
A1.1(b.0(a.1(b.1(b.0(y_0))))) → B1.0(b.1(b.0(y_0)))
A1.1(b.0(a.1(b.1(b.1(b.0(B.0(y_0))))))) → B1.1(b.1(b.1(b.1(b.0(B.0(y_0))))))
A1.1(b.0(a.1(b.1(b.1(b.0(B.1(y_0))))))) → B1.1(b.1(b.1(b.1(b.0(B.1(y_0))))))
A1.1(b.0(a.1(b.1(b.1(y_0))))) → B1.0(b.1(b.1(y_0)))
A1.1(b.0(a.1(b.1(b.1(b.0(B.0(y_0))))))) → B1.0(b.1(b.1(b.1(b.0(B.0(y_0))))))
A1.1(b.0(a.1(b.1(x0)))) → A1.0(b.0(a.0(a.0(a.1(x0)))))
B1.1(b.1(b.0(x))) → A1.1(b.0(x))
B1.1(b.1(b.1(x))) → A1.1(b.1(x))
A1.1(b.0(a.1(b.1(b.1(x0))))) → A1.0(b.0(a.1(b.1(x0))))
A1.1(b.0(a.1(b.0(y_0)))) → B1.1(b.1(b.0(y_0)))
A1.1(b.0(a.1(b.1(y_0)))) → B1.1(b.1(b.1(y_0)))
A1.1(b.0(a.1(b.0(x0)))) → A1.0(b.0(a.0(a.0(a.0(x0)))))
A1.1(b.0(a.1(b.1(y_0)))) → B1.0(b.1(b.1(y_0)))
A1.1(b.0(a.1(b.1(b.1(x0))))) → A1.1(b.0(a.1(b.1(x0))))
B1.1(b.1(b.1(x))) → A1.0(b.1(x))
A1.1(b.0(a.1(b.0(x0)))) → A1.1(b.0(a.0(a.0(a.0(x0)))))
A1.1(b.0(a.1(b.1(b.0(y_0))))) → B1.1(b.1(b.0(y_0)))
A1.1(b.0(a.1(b.1(b.0(x0))))) → A1.0(b.0(a.1(b.0(x0))))
A1.1(b.0(a.1(b.1(b.1(y_0))))) → B1.1(b.1(b.1(y_0)))

The TRS R consists of the following rules:

A.1(x0) → A.0(x0)
b.1(b.0(x)) → a.0(a.0(a.0(x)))
a.1(b.1(b.0(B.0(x)))) → a.1(b.1(b.0(A.0(x))))
b.0(B.0(x)) → A.0(x)
a.1(b.0(A.0(x))) → a.1(b.0(B.0(x)))
b.1(x0) → b.0(x0)
a.1(b.1(b.0(B.0(x)))) → a.1(b.1(b.0(a.0(A.0(x)))))
b.1(b.1(x)) → a.0(a.0(a.1(x)))
a.1(b.1(b.1(b.0(B.1(x))))) → a.1(b.1(b.0(B.1(x))))
a.1(x0) → a.0(x0)
a.1(b.0(a.0(x))) → a.1(b.1(b.0(x)))
b.1(b.1(b.1(x))) → a.1(b.1(x))
b.0(B.1(x)) → A.1(x)
b.1(b.1(b.0(x))) → a.1(b.0(x))
a.1(b.0(a.1(x))) → a.1(b.1(b.1(x)))
B.1(x0) → B.0(x0)
b.1(b.0(B.0(x))) → A.0(x)
a.1(b.1(b.1(b.0(B.0(x))))) → a.1(b.1(b.0(B.0(x))))
a.1(b.1(b.0(B.1(x)))) → a.1(b.1(b.0(A.1(x))))
a.1(b.0(A.1(x))) → a.1(b.0(B.1(x)))
b.1(b.0(B.1(x))) → A.1(x)
a.1(b.1(b.0(B.1(x)))) → a.1(b.1(b.0(a.0(A.1(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 14 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ QDPToSRSProof
                    ↳ QTRS
                      ↳ QTRS Reverse
                        ↳ QTRS
                          ↳ DependencyPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ ForwardInstantiation
                                                            ↳ QDP
                                                              ↳ ForwardInstantiation
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ ForwardInstantiation
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ ForwardInstantiation
                                                                                                                        ↳ QDP
                                                                                                                          ↳ SemLabProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ DependencyGraphProof
QDP
                                                                                                                                  ↳ RuleRemovalProof
                                                                                                                          ↳ SemLabProof2
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

A1.1(b.0(a.1(b.0(y_0)))) → B1.1(b.1(b.0(y_0)))
A1.1(b.0(a.1(b.1(y_0)))) → B1.1(b.1(b.1(y_0)))
A1.1(b.0(a.1(b.1(b.0(x0))))) → A1.1(b.0(a.1(b.0(x0))))
A1.1(b.0(a.1(b.1(b.1(x0))))) → A1.1(b.0(a.1(b.1(x0))))
A1.1(b.0(a.1(b.1(b.1(b.0(B.0(y_0))))))) → B1.1(b.1(b.1(b.1(b.0(B.0(y_0))))))
A1.1(b.0(a.1(b.1(b.1(b.0(B.1(y_0))))))) → B1.1(b.1(b.1(b.1(b.0(B.1(y_0))))))
A1.1(b.0(a.1(b.1(b.0(y_0))))) → B1.1(b.1(b.0(y_0)))
B1.1(b.1(b.1(x))) → A1.1(b.1(x))
B1.1(b.1(b.0(x))) → A1.1(b.0(x))
A1.1(b.0(a.1(b.1(b.1(y_0))))) → B1.1(b.1(b.1(y_0)))

The TRS R consists of the following rules:

A.1(x0) → A.0(x0)
b.1(b.0(x)) → a.0(a.0(a.0(x)))
a.1(b.1(b.0(B.0(x)))) → a.1(b.1(b.0(A.0(x))))
b.0(B.0(x)) → A.0(x)
a.1(b.0(A.0(x))) → a.1(b.0(B.0(x)))
b.1(x0) → b.0(x0)
a.1(b.1(b.0(B.0(x)))) → a.1(b.1(b.0(a.0(A.0(x)))))
b.1(b.1(x)) → a.0(a.0(a.1(x)))
a.1(b.1(b.1(b.0(B.1(x))))) → a.1(b.1(b.0(B.1(x))))
a.1(x0) → a.0(x0)
a.1(b.0(a.0(x))) → a.1(b.1(b.0(x)))
b.1(b.1(b.1(x))) → a.1(b.1(x))
b.0(B.1(x)) → A.1(x)
b.1(b.1(b.0(x))) → a.1(b.0(x))
a.1(b.0(a.1(x))) → a.1(b.1(b.1(x)))
B.1(x0) → B.0(x0)
b.1(b.0(B.0(x))) → A.0(x)
a.1(b.1(b.1(b.0(B.0(x))))) → a.1(b.1(b.0(B.0(x))))
a.1(b.1(b.0(B.1(x)))) → a.1(b.1(b.0(A.1(x))))
a.1(b.0(A.1(x))) → a.1(b.0(B.1(x)))
b.1(b.0(B.1(x))) → A.1(x)
a.1(b.1(b.0(B.1(x)))) → a.1(b.1(b.0(a.0(A.1(x)))))

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

Strictly oriented rules of the TRS R:

A.1(x0) → A.0(x0)
B.1(x0) → B.0(x0)

Used ordering: POLO with Polynomial interpretation [25]:

POL(A.0(x1)) = x1   
POL(A.1(x1)) = 1 + x1   
POL(A1.1(x1)) = x1   
POL(B.0(x1)) = x1   
POL(B.1(x1)) = 1 + x1   
POL(B1.1(x1)) = x1   
POL(a.0(x1)) = x1   
POL(a.1(x1)) = x1   
POL(b.0(x1)) = x1   
POL(b.1(x1)) = x1   



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ QDPToSRSProof
                    ↳ QTRS
                      ↳ QTRS Reverse
                        ↳ QTRS
                          ↳ DependencyPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ ForwardInstantiation
                                                            ↳ QDP
                                                              ↳ ForwardInstantiation
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ ForwardInstantiation
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ ForwardInstantiation
                                                                                                                        ↳ QDP
                                                                                                                          ↳ SemLabProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ DependencyGraphProof
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ RuleRemovalProof
QDP
                                                                                                                          ↳ SemLabProof2
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

A1.1(b.0(a.1(b.0(y_0)))) → B1.1(b.1(b.0(y_0)))
A1.1(b.0(a.1(b.1(y_0)))) → B1.1(b.1(b.1(y_0)))
A1.1(b.0(a.1(b.1(b.0(x0))))) → A1.1(b.0(a.1(b.0(x0))))
A1.1(b.0(a.1(b.1(b.1(x0))))) → A1.1(b.0(a.1(b.1(x0))))
A1.1(b.0(a.1(b.1(b.1(b.0(B.0(y_0))))))) → B1.1(b.1(b.1(b.1(b.0(B.0(y_0))))))
A1.1(b.0(a.1(b.1(b.1(b.0(B.1(y_0))))))) → B1.1(b.1(b.1(b.1(b.0(B.1(y_0))))))
A1.1(b.0(a.1(b.1(b.0(y_0))))) → B1.1(b.1(b.0(y_0)))
B1.1(b.1(b.0(x))) → A1.1(b.0(x))
B1.1(b.1(b.1(x))) → A1.1(b.1(x))
A1.1(b.0(a.1(b.1(b.1(y_0))))) → B1.1(b.1(b.1(y_0)))

The TRS R consists of the following rules:

b.1(b.0(x)) → a.0(a.0(a.0(x)))
a.1(b.1(b.0(B.0(x)))) → a.1(b.1(b.0(A.0(x))))
b.0(B.0(x)) → A.0(x)
a.1(b.0(A.0(x))) → a.1(b.0(B.0(x)))
b.1(x0) → b.0(x0)
a.1(b.1(b.0(B.0(x)))) → a.1(b.1(b.0(a.0(A.0(x)))))
b.1(b.1(x)) → a.0(a.0(a.1(x)))
a.1(b.1(b.1(b.0(B.1(x))))) → a.1(b.1(b.0(B.1(x))))
a.1(x0) → a.0(x0)
a.1(b.0(a.0(x))) → a.1(b.1(b.0(x)))
b.1(b.1(b.1(x))) → a.1(b.1(x))
b.0(B.1(x)) → A.1(x)
b.1(b.1(b.0(x))) → a.1(b.0(x))
a.1(b.0(a.1(x))) → a.1(b.1(b.1(x)))
b.1(b.0(B.0(x))) → A.0(x)
a.1(b.1(b.1(b.0(B.0(x))))) → a.1(b.1(b.0(B.0(x))))
a.1(b.1(b.0(B.1(x)))) → a.1(b.1(b.0(A.1(x))))
a.1(b.0(A.1(x))) → a.1(b.0(B.1(x)))
b.1(b.0(B.1(x))) → A.1(x)
a.1(b.1(b.0(B.1(x)))) → a.1(b.1(b.0(a.0(A.1(x)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
As can be seen after transforming the QDP problem by semantic labelling [33] and then some rule deleting processors, only certain labelled rules and pairs can be used. Hence, we only have to consider all unlabelled pairs and rules (without the decreasing rules for quasi-models).

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ QDPToSRSProof
                    ↳ QTRS
                      ↳ QTRS Reverse
                        ↳ QTRS
                          ↳ DependencyPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ ForwardInstantiation
                                                            ↳ QDP
                                                              ↳ ForwardInstantiation
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ ForwardInstantiation
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ ForwardInstantiation
                                                                                                                        ↳ QDP
                                                                                                                          ↳ SemLabProof
                                                                                                                          ↳ SemLabProof2
QDP
                          ↳ QTRS Reverse
                          ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

A1(b(a(b(b(b(B(y_0))))))) → B1(b(b(b(b(B(y_0))))))
A1(b(a(b(y_0)))) → B1(b(b(y_0)))
B1(b(b(x))) → A1(b(x))
A1(b(a(b(b(x0))))) → A1(b(a(b(x0))))
A1(b(a(b(b(y_0))))) → B1(b(b(y_0)))

The TRS R consists of the following rules:

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

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

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

The set Q is empty.

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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

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

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

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

Q is empty.

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

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

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

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

Q is empty.

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

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

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

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

Q is empty.