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

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

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

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

Q is empty.

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

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

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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

A(c(x0)) → B(x0)
A(c(c(x0))) → B(c(c(c(x0))))
A(c(a(x0))) → B(a(a(a(x0))))



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

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

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

The TRS R consists of the following rules:

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

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

A(b(c(y0))) → B(c(c(y0)))



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

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

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

The TRS R consists of the following rules:

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

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

A(c(c(y0))) → B(c(c(y0)))



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

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

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

The TRS R consists of the following rules:

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

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

A(b(c(y0))) → B(c(y0))



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

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

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

The TRS R consists of the following rules:

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

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

A(c(c(y0))) → B(c(y0))



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

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

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

The TRS R consists of the following rules:

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

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

A(b(c(x0))) → B(x0)



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

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

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

The TRS R consists of the following rules:

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

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

A(c(c(x0))) → B(x0)



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

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

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

The TRS R consists of the following rules:

a(b(x1)) → b(b(b(x1)))
a(c(x1)) → c(b(b(x1)))
b(a(x1)) → a(a(a(x1)))
b(c(x1)) → c(c(c(x1)))
a(x1) → x1
b(x1) → x1
c(x1) → 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
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ 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(x1)) → b(b(b(x1)))
a(c(x1)) → c(b(b(x1)))
b(a(x1)) → a(a(a(x1)))
b(c(x1)) → c(c(c(x1)))
a(x1) → x1
b(x1) → x1
c(x1) → x1
A(c(c(x0))) → B(x0)
A(b(x0)) → B(x0)
A(b(c(x0))) → B(x0)
A(c(a(x0))) → B(a(a(a(x0))))
A(b(c(x0))) → B(b(c(c(c(x0)))))
B(a(x0)) → A(x0)
A(c(x1)) → B(x1)
B(a(c(x0))) → A(a(c(b(b(x0)))))
B(a(c(x0))) → A(c(b(b(x0))))
A(b(a(x0))) → B(a(a(a(x0))))
B(a(b(x0))) → A(b(b(b(x0))))
A(b(a(x0))) → B(b(a(a(a(x0)))))
B(a(b(x0))) → A(a(b(b(b(x0)))))

Q is empty.

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

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

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

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

The set Q is empty.

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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

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

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

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

Q is empty.

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

b(a(x)) → b(b(b(x)))
c(a(x)) → b(b(c(x)))
a(b(x)) → a(a(a(x)))
c(b(x)) → c(c(c(x)))
a(x) → x
b(x) → x
c(x) → x
c(c(A(x))) → B(x)
b(A(x)) → B(x)
c(b(A(x))) → B(x)
a(c(A(x))) → a(a(a(B(x))))
c(b(A(x))) → c(c(c(b(B(x)))))
a(B(x)) → A(x)
c(A(x)) → B(x)
c(a(B(x))) → b(b(c(a(A(x)))))
c(a(B(x))) → b(b(c(A(x))))
a(b(A(x))) → a(a(a(B(x))))
b(a(B(x))) → b(b(b(A(x))))
a(b(A(x))) → a(a(a(b(B(x)))))
b(a(B(x))) → b(b(b(a(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 3 SCCs with 14 less nodes.

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

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ UsableRulesReductionPairsProof
                                                                        ↳ UsableRulesProof
                                                                      ↳ QDP
                                                                      ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

A1(c(A(x))) → A1(a(a(B(x))))
A1(c(A(x))) → A1(a(B(x)))
The following rules are removed from R:

a(c(A(x))) → a(a(a(B(x))))
Used ordering: POLO with Polynomial interpretation [25]:

POL(A(x1)) = x1   
POL(A1(x1)) = x1   
POL(B(x1)) = x1   
POL(a(x1)) = x1   
POL(b(x1)) = 2·x1   
POL(c(x1)) = 1 + 2·x1   



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ UsableRulesReductionPairsProof
                                                                              ↳ QDP
                                                                                ↳ RuleRemovalProof
                                                                        ↳ UsableRulesProof
                                                                      ↳ QDP
                                                                      ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

Strictly oriented rules of the TRS R:

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

Used ordering: POLO with Polynomial interpretation [25]:

POL(A(x1)) = 2 + 2·x1   
POL(A1(x1)) = x1   
POL(B(x1)) = 1 + 2·x1   
POL(a(x1)) = 1 + x1   
POL(b(x1)) = 2 + 2·x1   



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ UsableRulesReductionPairsProof
                                                                              ↳ QDP
                                                                                ↳ RuleRemovalProof
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof
                                                                        ↳ UsableRulesProof
                                                                      ↳ QDP
                                                                      ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

a(b(x)) → a(a(a(x)))
a(b(A(x))) → a(a(a(b(B(x)))))
a(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 0 SCCs with 2 less nodes.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                        ↳ UsableRulesProof
                                                                      ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

Strictly oriented rules of the TRS R:

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

Used ordering: POLO with Polynomial interpretation [25]:

POL(A(x1)) = 1 + x1   
POL(B(x1)) = 2 + x1   
POL(B1(x1)) = x1   
POL(a(x1)) = 2 + 2·x1   
POL(b(x1)) = 1 + x1   



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                        ↳ UsableRulesProof
                                                                      ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

c(a(x)) → b(b(c(x)))
c(b(A(x))) → c(c(c(b(B(x)))))
c(b(x)) → c(c(c(x)))
c(x) → x
c(c(A(x))) → B(x)
c(b(A(x))) → B(x)
c(A(x)) → B(x)
c(a(B(x))) → b(b(c(a(A(x)))))
c(a(B(x))) → b(b(c(A(x))))
b(x) → x
b(A(x)) → B(x)
b(a(B(x))) → b(b(b(a(A(x)))))
b(a(x)) → b(b(b(x)))
b(a(B(x))) → b(b(b(A(x))))
a(x) → 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 dependency pairs:

C(a(x)) → C(x)

Strictly oriented rules of the TRS R:

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

Used ordering: POLO with Polynomial interpretation [25]:

POL(A(x1)) = x1   
POL(B(x1)) = x1   
POL(C(x1)) = 2·x1   
POL(a(x1)) = 1 + x1   
POL(b(x1)) = x1   
POL(c(x1)) = x1   



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

c(b(A(x))) → c(c(c(b(B(x)))))
c(b(x)) → c(c(c(x)))
c(x) → x
c(c(A(x))) → B(x)
c(b(A(x))) → B(x)
c(A(x)) → B(x)
c(a(B(x))) → b(b(c(a(A(x)))))
b(x) → x
b(A(x)) → B(x)
b(a(B(x))) → b(b(b(a(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
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C(b(A(x))) → C(b(B(x)))
C(b(x)) → C(c(c(x)))
C(b(A(x))) → C(c(c(b(B(x)))))
C(b(x)) → C(c(x))
C(b(A(x))) → C(c(b(B(x))))
C(b(x)) → C(x)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


C(b(A(x))) → C(b(B(x)))
C(b(A(x))) → C(c(c(b(B(x)))))
C(b(A(x))) → C(c(b(B(x))))
The remaining pairs can at least be oriented weakly.

C(b(x)) → C(c(c(x)))
C(b(x)) → C(c(x))
C(b(x)) → C(x)
Used ordering: Polynomial Order [21,25] with Interpretation:

POL( A(x1) ) = x1 + 1


POL( C(x1) ) = x1


POL( c(x1) ) = x1


POL( b(x1) ) = x1


POL( B(x1) ) = max{0, x1 - 1}


POL( a(x1) ) = max{0, -1}



The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C(b(x)) → C(c(c(x)))
C(b(x)) → C(c(x))
C(b(x)) → C(x)

The TRS R consists of the following rules:

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

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

C(b(A(x0))) → C(B(x0))
C(b(b(x0))) → C(c(c(c(c(x0)))))
C(b(A(x0))) → C(c(B(x0)))
C(b(a(B(x0)))) → C(c(b(b(c(a(A(x0)))))))
C(b(y0)) → C(c(y0))
C(b(b(A(x0)))) → C(c(c(c(c(b(B(x0)))))))
C(b(b(A(x0)))) → C(c(B(x0)))
C(b(c(A(x0)))) → C(c(B(x0)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C(b(b(x0))) → C(c(c(c(c(x0)))))
C(b(A(x0))) → C(B(x0))
C(b(A(x0))) → C(c(B(x0)))
C(b(a(B(x0)))) → C(c(b(b(c(a(A(x0)))))))
C(b(x)) → C(c(x))
C(b(b(A(x0)))) → C(c(c(c(c(b(B(x0)))))))
C(b(b(A(x0)))) → C(c(B(x0)))
C(b(x)) → C(x)
C(b(c(A(x0)))) → C(c(B(x0)))

The TRS R consists of the following rules:

c(b(A(x))) → c(c(c(b(B(x)))))
c(b(x)) → c(c(c(x)))
c(x) → x
c(c(A(x))) → B(x)
c(b(A(x))) → B(x)
c(A(x)) → B(x)
c(a(B(x))) → b(b(c(a(A(x)))))
b(x) → x
b(A(x)) → B(x)
b(a(B(x))) → b(b(b(a(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
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C(b(b(x0))) → C(c(c(c(c(x0)))))
C(b(A(x0))) → C(c(B(x0)))
C(b(a(B(x0)))) → C(c(b(b(c(a(A(x0)))))))
C(b(x)) → C(c(x))
C(b(b(A(x0)))) → C(c(c(c(c(b(B(x0)))))))
C(b(b(A(x0)))) → C(c(B(x0)))
C(b(x)) → C(x)
C(b(c(A(x0)))) → C(c(B(x0)))

The TRS R consists of the following rules:

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

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C(b(c(A(x0)))) → C(B(x0))
C(b(b(x0))) → C(c(c(c(c(x0)))))
C(b(A(x0))) → C(B(x0))
C(b(b(A(x0)))) → C(c(c(c(b(B(x0))))))
C(b(a(B(x0)))) → C(c(b(b(c(a(A(x0)))))))
C(b(a(B(x0)))) → C(b(b(c(a(A(x0))))))
C(b(x)) → C(x)
C(b(A(x0))) → C(c(B(x0)))
C(b(b(A(x0)))) → C(B(x0))
C(b(b(A(x0)))) → C(c(c(c(c(b(B(x0)))))))
C(b(b(x0))) → C(c(c(c(x0))))
C(b(b(A(x0)))) → C(c(B(x0)))
C(b(c(A(x0)))) → C(c(B(x0)))

The TRS R consists of the following rules:

c(b(A(x))) → c(c(c(b(B(x)))))
c(b(x)) → c(c(c(x)))
c(x) → x
c(c(A(x))) → B(x)
c(b(A(x))) → B(x)
c(A(x)) → B(x)
c(a(B(x))) → b(b(c(a(A(x)))))
b(x) → x
b(A(x)) → B(x)
b(a(B(x))) → b(b(b(a(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
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C(b(b(x0))) → C(c(c(c(c(x0)))))
C(b(b(A(x0)))) → C(c(c(c(b(B(x0))))))
C(b(A(x0))) → C(c(B(x0)))
C(b(a(B(x0)))) → C(c(b(b(c(a(A(x0)))))))
C(b(a(B(x0)))) → C(b(b(c(a(A(x0))))))
C(b(b(A(x0)))) → C(c(c(c(c(b(B(x0)))))))
C(b(b(x0))) → C(c(c(c(x0))))
C(b(b(A(x0)))) → C(c(B(x0)))
C(b(x)) → C(x)
C(b(c(A(x0)))) → C(c(B(x0)))

The TRS R consists of the following rules:

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

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

C(b(A(y0))) → C(B(y0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ DependencyGraphProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C(b(A(y0))) → C(B(y0))
C(b(b(x0))) → C(c(c(c(c(x0)))))
C(b(b(A(x0)))) → C(c(c(c(b(B(x0))))))
C(b(a(B(x0)))) → C(c(b(b(c(a(A(x0)))))))
C(b(b(A(x0)))) → C(c(c(c(c(b(B(x0)))))))
C(b(a(B(x0)))) → C(b(b(c(a(A(x0))))))
C(b(b(x0))) → C(c(c(c(x0))))
C(b(x)) → C(x)
C(b(b(A(x0)))) → C(c(B(x0)))
C(b(c(A(x0)))) → C(c(B(x0)))

The TRS R consists of the following rules:

c(b(A(x))) → c(c(c(b(B(x)))))
c(b(x)) → c(c(c(x)))
c(x) → x
c(c(A(x))) → B(x)
c(b(A(x))) → B(x)
c(A(x)) → B(x)
c(a(B(x))) → b(b(c(a(A(x)))))
b(x) → x
b(A(x)) → B(x)
b(a(B(x))) → b(b(b(a(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
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ DependencyGraphProof
                                                                                                              ↳ QDP
                                                                                                                ↳ Narrowing
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C(b(b(x0))) → C(c(c(c(c(x0)))))
C(b(b(A(x0)))) → C(c(c(c(b(B(x0))))))
C(b(a(B(x0)))) → C(c(b(b(c(a(A(x0)))))))
C(b(a(B(x0)))) → C(b(b(c(a(A(x0))))))
C(b(b(A(x0)))) → C(c(c(c(c(b(B(x0)))))))
C(b(b(x0))) → C(c(c(c(x0))))
C(b(b(A(x0)))) → C(c(B(x0)))
C(b(x)) → C(x)
C(b(c(A(x0)))) → C(c(B(x0)))

The TRS R consists of the following rules:

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

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

C(b(b(A(y0)))) → C(B(y0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ DependencyGraphProof
                                                                                                              ↳ QDP
                                                                                                                ↳ Narrowing
                                                                                                                  ↳ QDP
                                                                                                                    ↳ DependencyGraphProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C(b(b(x0))) → C(c(c(c(c(x0)))))
C(b(b(A(x0)))) → C(c(c(c(b(B(x0))))))
C(b(b(A(y0)))) → C(B(y0))
C(b(a(B(x0)))) → C(c(b(b(c(a(A(x0)))))))
C(b(b(A(x0)))) → C(c(c(c(c(b(B(x0)))))))
C(b(a(B(x0)))) → C(b(b(c(a(A(x0))))))
C(b(b(x0))) → C(c(c(c(x0))))
C(b(x)) → C(x)
C(b(c(A(x0)))) → C(c(B(x0)))

The TRS R consists of the following rules:

c(b(A(x))) → c(c(c(b(B(x)))))
c(b(x)) → c(c(c(x)))
c(x) → x
c(c(A(x))) → B(x)
c(b(A(x))) → B(x)
c(A(x)) → B(x)
c(a(B(x))) → b(b(c(a(A(x)))))
b(x) → x
b(A(x)) → B(x)
b(a(B(x))) → b(b(b(a(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
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ DependencyGraphProof
                                                                                                              ↳ QDP
                                                                                                                ↳ Narrowing
                                                                                                                  ↳ QDP
                                                                                                                    ↳ DependencyGraphProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Narrowing
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C(b(b(x0))) → C(c(c(c(c(x0)))))
C(b(b(A(x0)))) → C(c(c(c(b(B(x0))))))
C(b(a(B(x0)))) → C(c(b(b(c(a(A(x0)))))))
C(b(a(B(x0)))) → C(b(b(c(a(A(x0))))))
C(b(b(A(x0)))) → C(c(c(c(c(b(B(x0)))))))
C(b(b(x0))) → C(c(c(c(x0))))
C(b(x)) → C(x)
C(b(c(A(x0)))) → C(c(B(x0)))

The TRS R consists of the following rules:

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

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

C(b(c(A(y0)))) → C(B(y0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ DependencyGraphProof
                                                                                                              ↳ QDP
                                                                                                                ↳ Narrowing
                                                                                                                  ↳ QDP
                                                                                                                    ↳ DependencyGraphProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Narrowing
                                                                                                                          ↳ QDP
                                                                                                                            ↳ DependencyGraphProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C(b(c(A(y0)))) → C(B(y0))
C(b(b(x0))) → C(c(c(c(c(x0)))))
C(b(b(A(x0)))) → C(c(c(c(b(B(x0))))))
C(b(a(B(x0)))) → C(c(b(b(c(a(A(x0)))))))
C(b(b(A(x0)))) → C(c(c(c(c(b(B(x0)))))))
C(b(a(B(x0)))) → C(b(b(c(a(A(x0))))))
C(b(b(x0))) → C(c(c(c(x0))))
C(b(x)) → C(x)

The TRS R consists of the following rules:

c(b(A(x))) → c(c(c(b(B(x)))))
c(b(x)) → c(c(c(x)))
c(x) → x
c(c(A(x))) → B(x)
c(b(A(x))) → B(x)
c(A(x)) → B(x)
c(a(B(x))) → b(b(c(a(A(x)))))
b(x) → x
b(A(x)) → B(x)
b(a(B(x))) → b(b(b(a(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
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ DependencyGraphProof
                                                                                                              ↳ QDP
                                                                                                                ↳ Narrowing
                                                                                                                  ↳ QDP
                                                                                                                    ↳ DependencyGraphProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Narrowing
                                                                                                                          ↳ QDP
                                                                                                                            ↳ DependencyGraphProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C(b(b(x0))) → C(c(c(c(c(x0)))))
C(b(b(A(x0)))) → C(c(c(c(b(B(x0))))))
C(b(a(B(x0)))) → C(c(b(b(c(a(A(x0)))))))
C(b(a(B(x0)))) → C(b(b(c(a(A(x0))))))
C(b(b(A(x0)))) → C(c(c(c(c(b(B(x0)))))))
C(b(b(x0))) → C(c(c(c(x0))))
C(b(x)) → C(x)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


C(b(b(A(x0)))) → C(c(c(c(b(B(x0))))))
C(b(b(A(x0)))) → C(c(c(c(c(b(B(x0)))))))
The remaining pairs can at least be oriented weakly.

C(b(b(x0))) → C(c(c(c(c(x0)))))
C(b(a(B(x0)))) → C(c(b(b(c(a(A(x0)))))))
C(b(a(B(x0)))) → C(b(b(c(a(A(x0))))))
C(b(b(x0))) → C(c(c(c(x0))))
C(b(x)) → C(x)
Used ordering: Polynomial interpretation [25,35]:

POL(C(x1)) = (4)x_1   
POL(c(x1)) = (2)x_1   
POL(B(x1)) = 0   
POL(a(x1)) = 0   
POL(A(x1)) = 4   
POL(b(x1)) = (4)x_1   
The value of delta used in the strict ordering is 256.
The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ DependencyGraphProof
                                                                                                              ↳ QDP
                                                                                                                ↳ Narrowing
                                                                                                                  ↳ QDP
                                                                                                                    ↳ DependencyGraphProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Narrowing
                                                                                                                          ↳ QDP
                                                                                                                            ↳ DependencyGraphProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ SemLabProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C(b(b(x0))) → C(c(c(c(c(x0)))))
C(b(a(B(x0)))) → C(c(b(b(c(a(A(x0)))))))
C(b(a(B(x0)))) → C(b(b(c(a(A(x0))))))
C(b(b(x0))) → C(c(c(c(x0))))
C(b(x)) → C(x)

The TRS R consists of the following rules:

c(b(A(x))) → c(c(c(b(B(x)))))
c(b(x)) → c(c(c(x)))
c(x) → x
c(c(A(x))) → B(x)
c(b(A(x))) → B(x)
c(A(x)) → B(x)
c(a(B(x))) → b(b(c(a(A(x)))))
b(x) → x
b(A(x)) → B(x)
b(a(B(x))) → b(b(b(a(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.C: 0
c: 1
B: 1
a: 0
A: 0
b: 1
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ DependencyGraphProof
                                                                                                              ↳ QDP
                                                                                                                ↳ Narrowing
                                                                                                                  ↳ QDP
                                                                                                                    ↳ DependencyGraphProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Narrowing
                                                                                                                          ↳ QDP
                                                                                                                            ↳ DependencyGraphProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ SemLabProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ DependencyGraphProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

b.0(A.1(x)) → B.1(x)
c.1(x) → x
b.0(a.1(B.0(x))) → b.1(b.1(b.0(a.0(A.0(x)))))
A.1(x0) → A.0(x0)
c.1(c.0(A.1(x))) → B.1(x)
b.1(x0) → b.0(x0)
a.1(x0) → a.0(x0)
c.1(b.0(A.1(x))) → c.1(c.1(c.1(b.1(B.1(x)))))
B.1(x0) → B.0(x0)
c.0(x) → x
c.1(c.0(A.0(x))) → B.0(x)
b.1(x) → x
c.1(x0) → c.0(x0)
c.0(a.1(B.1(x))) → b.1(b.1(c.0(a.0(A.1(x)))))
b.0(A.0(x)) → B.0(x)
c.1(b.0(A.1(x))) → B.1(x)
c.0(A.1(x)) → B.1(x)
c.1(b.0(A.0(x))) → B.0(x)
c.1(b.0(A.0(x))) → c.1(c.1(c.1(b.1(B.0(x)))))
c.0(A.0(x)) → B.0(x)
c.0(a.1(B.0(x))) → b.1(b.1(c.0(a.0(A.0(x)))))
b.0(a.1(B.1(x))) → b.1(b.1(b.0(a.0(A.1(x)))))
c.1(b.1(x)) → c.1(c.1(c.1(x)))
c.1(b.0(x)) → c.1(c.1(c.0(x)))
b.0(x) → 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 10 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ DependencyGraphProof
                                                                                                              ↳ QDP
                                                                                                                ↳ Narrowing
                                                                                                                  ↳ QDP
                                                                                                                    ↳ DependencyGraphProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Narrowing
                                                                                                                          ↳ QDP
                                                                                                                            ↳ DependencyGraphProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ SemLabProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ UsableRulesReductionPairsProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

C.1(b.0(a.1(B.0(x0)))) → C.1(c.1(b.1(b.1(c.0(a.0(A.0(x0)))))))
C.1(b.0(a.1(B.0(x0)))) → C.1(b.1(b.1(c.0(a.0(A.0(x0))))))
C.1(b.0(a.1(B.1(x0)))) → C.1(b.1(b.1(c.0(a.0(A.1(x0))))))
C.1(b.0(a.1(B.1(x0)))) → C.1(c.1(b.1(b.1(c.0(a.0(A.1(x0)))))))
The following rules are removed from R:

c.0(a.1(B.1(x))) → b.1(b.1(c.0(a.0(A.1(x)))))
c.0(a.1(B.0(x))) → b.1(b.1(c.0(a.0(A.0(x)))))
b.0(a.1(B.1(x))) → b.1(b.1(b.0(a.0(A.1(x)))))
b.0(a.1(B.0(x))) → b.1(b.1(b.0(a.0(A.0(x)))))
Used ordering: POLO with Polynomial interpretation [25]:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ DependencyGraphProof
                                                                                                              ↳ QDP
                                                                                                                ↳ Narrowing
                                                                                                                  ↳ QDP
                                                                                                                    ↳ DependencyGraphProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Narrowing
                                                                                                                          ↳ QDP
                                                                                                                            ↳ DependencyGraphProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ SemLabProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ UsableRulesReductionPairsProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ UsableRulesReductionPairsProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C.1(b.1(b.1(x0))) → C.1(c.1(c.1(c.1(c.1(x0)))))
C.1(b.1(b.0(x0))) → C.1(c.1(c.1(c.0(x0))))
C.1(b.1(b.0(x0))) → C.1(c.1(c.1(c.1(c.0(x0)))))
C.1(b.1(x)) → C.1(x)
C.1(b.1(b.1(x0))) → C.1(c.1(c.1(c.1(x0))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

C.1(b.1(b.1(x0))) → C.1(c.1(c.1(c.1(c.1(x0)))))
C.1(b.1(b.0(x0))) → C.1(c.1(c.1(c.0(x0))))
C.1(b.1(b.0(x0))) → C.1(c.1(c.1(c.1(c.0(x0)))))
C.1(b.1(x)) → C.1(x)
C.1(b.1(b.1(x0))) → C.1(c.1(c.1(c.1(x0))))
The following rules are removed from R:

c.1(c.0(A.1(x))) → B.1(x)
c.1(b.0(A.1(x))) → c.1(c.1(c.1(b.1(B.1(x)))))
c.1(b.1(x)) → c.1(c.1(c.1(x)))
c.1(b.0(A.0(x))) → c.1(c.1(c.1(b.1(B.0(x)))))
c.1(b.0(x)) → c.1(c.1(c.0(x)))
c.1(c.0(A.0(x))) → B.0(x)
c.1(b.0(A.1(x))) → B.1(x)
c.1(b.0(A.0(x))) → B.0(x)
c.0(A.1(x)) → B.1(x)
c.0(A.0(x)) → B.0(x)
b.1(x) → x
b.0(x) → x
b.0(A.1(x)) → B.1(x)
b.0(A.0(x)) → B.0(x)
Used ordering: POLO with Polynomial interpretation [25]:

POL(A.0(x1)) = 1 + x1   
POL(A.1(x1)) = 1 + x1   
POL(B.0(x1)) = x1   
POL(B.1(x1)) = x1   
POL(C.1(x1)) = x1   
POL(b.0(x1)) = 1 + x1   
POL(b.1(x1)) = 1 + x1   
POL(c.0(x1)) = x1   
POL(c.1(x1)) = x1   



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ QDPToSRSProof
                                                        ↳ QTRS
                                                          ↳ QTRS Reverse
                                                            ↳ QTRS
                                                              ↳ QTRS Reverse
                                                              ↳ QTRS Reverse
                                                              ↳ DependencyPairsProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ AND
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                      ↳ QDP
                                                                        ↳ UsableRulesProof
                                                                          ↳ QDP
                                                                            ↳ RuleRemovalProof
                                                                              ↳ QDP
                                                                                ↳ DependencyGraphProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ DependencyGraphProof
                                                                                                              ↳ QDP
                                                                                                                ↳ Narrowing
                                                                                                                  ↳ QDP
                                                                                                                    ↳ DependencyGraphProof
                                                                                                                      ↳ QDP
                                                                                                                        ↳ Narrowing
                                                                                                                          ↳ QDP
                                                                                                                            ↳ DependencyGraphProof
                                                                                                                              ↳ QDP
                                                                                                                                ↳ QDPOrderProof
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ SemLabProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ DependencyGraphProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ UsableRulesReductionPairsProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ UsableRulesReductionPairsProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ PisEmptyProof
                                                                        ↳ UsableRulesProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

Q DP problem:
P is empty.
The TRS R consists of the following rules:

c.1(x) → x
c.1(x0) → c.0(x0)
B.1(x0) → B.0(x0)
c.0(x) → x
b.1(x0) → b.0(x0)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

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

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

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

The TRS R consists of the following rules:

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

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

b(a(x)) → b(b(b(x)))
c(a(x)) → b(b(c(x)))
a(b(x)) → a(a(a(x)))
c(b(x)) → c(c(c(x)))
a(x) → x
b(x) → x
c(x) → 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:

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

Q is empty.

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

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

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

b(a(x)) → b(b(b(x)))
c(a(x)) → b(b(c(x)))
a(b(x)) → a(a(a(x)))
c(b(x)) → c(c(c(x)))
a(x) → x
b(x) → x
c(x) → 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:

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

Q is empty.