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

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

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

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

Q is empty.

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

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

The TRS R consists of the following rules:

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

C(c(b(x0))) → C(c(c(c(c(x0)))))



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

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

B(b(x1)) → C(c(x1))
B(b(x1)) → C(c(c(x1)))
B(b(x1)) → C(c(c(c(x1))))
B(b(x1)) → C(x1)
C(c(x1)) → B(x1)
C(c(b(x0))) → C(c(c(c(c(x0)))))

The TRS R consists of the following rules:

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

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


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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

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

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

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

C1(c(x)) → A(x)
B1(B(x)) → C1(c(c(C(x))))
B1(b(x)) → C1(c(c(c(x))))
B1(c(C(x))) → C1(c(C(x)))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(C(x))))
B1(b(x)) → C1(c(x))
B1(b(x)) → C1(c(c(x)))
C1(c(x)) → B1(c(a(x)))
B1(B(x)) → C1(c(C(x)))
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(B(x)) → C1(C(x))
C1(c(x)) → C1(a(x))

The TRS R consists of the following rules:

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

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

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

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

C1(c(x)) → A(x)
B1(B(x)) → C1(c(c(C(x))))
B1(b(x)) → C1(c(c(c(x))))
B1(c(C(x))) → C1(c(C(x)))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(C(x))))
B1(b(x)) → C1(c(x))
B1(b(x)) → C1(c(c(x)))
C1(c(x)) → B1(c(a(x)))
B1(B(x)) → C1(c(C(x)))
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(B(x)) → C1(C(x))
C1(c(x)) → C1(a(x))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

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

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

B1(B(x)) → C1(c(c(C(x))))
B1(b(x)) → C1(c(c(x)))
B1(b(x)) → C1(c(c(c(x))))
C1(c(x)) → B1(c(a(x)))
B1(B(x)) → C1(c(C(x)))
B1(c(C(x))) → C1(c(C(x)))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(c(C(x))) → C1(c(c(C(x))))
B1(b(x)) → C1(c(x))
C1(c(x)) → C1(a(x))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(x)) → B1(c(a(x))) at position [0] we obtained the following new rules:

C1(c(a(x0))) → B1(c(x0))



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

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

B1(B(x)) → C1(c(c(C(x))))
B1(b(x)) → C1(c(c(c(x))))
B1(b(x)) → C1(c(c(x)))
B1(B(x)) → C1(c(C(x)))
B1(c(C(x))) → C1(c(C(x)))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(c(C(x))) → C1(c(c(C(x))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → B1(c(x0))
C1(c(x)) → C1(a(x))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(x)) → C1(a(x)) at position [0] we obtained the following new rules:

C1(c(a(x0))) → C1(x0)



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

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

B1(B(x)) → C1(c(c(C(x))))
B1(b(x)) → C1(c(c(x)))
B1(b(x)) → C1(c(c(c(x))))
B1(B(x)) → C1(c(C(x)))
B1(c(C(x))) → C1(c(C(x)))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(c(C(x))) → C1(c(c(C(x))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(B(x)) → C1(c(C(x))) at position [0] we obtained the following new rules:

B1(B(x0)) → C1(B(x0))



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

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

B1(B(x0)) → C1(B(x0))
B1(B(x)) → C1(c(c(C(x))))
B1(b(x)) → C1(c(c(c(x))))
B1(b(x)) → C1(c(c(x)))
B1(c(C(x))) → C1(c(C(x)))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(c(C(x))) → C1(c(c(C(x))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))

The TRS R consists of the following rules:

a(a(x)) → x
b(b(x)) → c(c(c(c(x))))
c(c(x)) → b(c(a(x)))
b(B(x)) → c(C(x))
b(B(x)) → c(c(C(x)))
b(B(x)) → c(c(c(C(x))))
b(B(x)) → C(x)
c(C(x)) → B(x)
b(c(C(x))) → c(c(c(c(C(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
              ↳ QDPToSRSProof
                ↳ QTRS
                  ↳ QTRS Reverse
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                      ↳ QTRS Reverse
                      ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(B(x)) → C1(c(c(C(x))))
B1(b(x)) → C1(c(c(x)))
B1(b(x)) → C1(c(c(c(x))))
B1(c(C(x))) → C1(c(C(x)))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(c(C(x))) → C1(c(c(C(x))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(B(x)) → C1(c(c(C(x)))) at position [0] we obtained the following new rules:

B1(B(x0)) → C1(c(B(x0)))
B1(B(y0)) → C1(b(c(a(C(y0)))))



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

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

B1(b(x)) → C1(c(c(c(x))))
B1(b(x)) → C1(c(c(x)))
B1(c(C(x))) → C1(c(C(x)))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(B(x0)) → C1(c(B(x0)))
B1(c(C(x))) → C1(c(c(C(x))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))
B1(B(y0)) → C1(b(c(a(C(y0)))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

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

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

B1(b(x)) → C1(c(c(x)))
B1(b(x)) → C1(c(c(c(x))))
B1(c(C(x))) → C1(c(C(x)))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(c(C(x))) → C1(c(c(C(x))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(c(C(x))) → C1(c(c(C(x)))) at position [0] we obtained the following new rules:

B1(c(C(y0))) → C1(b(c(a(C(y0)))))
B1(c(C(x0))) → C1(c(B(x0)))



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

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

B1(b(x)) → C1(c(c(c(x))))
B1(b(x)) → C1(c(c(x)))
B1(c(C(y0))) → C1(b(c(a(C(y0)))))
B1(c(C(x0))) → C1(c(B(x0)))
B1(c(C(x))) → C1(c(C(x)))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

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

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

B1(b(x)) → C1(c(c(x)))
B1(b(x)) → C1(c(c(c(x))))
B1(c(C(x))) → C1(c(C(x)))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(c(C(x))) → C1(c(C(x))) at position [0] we obtained the following new rules:

B1(c(C(x0))) → C1(B(x0))



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

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

B1(b(x)) → C1(c(c(c(x))))
B1(b(x)) → C1(c(c(x)))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(c(C(x0))) → C1(B(x0))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))

The TRS R consists of the following rules:

a(a(x)) → x
b(b(x)) → c(c(c(c(x))))
c(c(x)) → b(c(a(x)))
b(B(x)) → c(C(x))
b(B(x)) → c(c(C(x)))
b(B(x)) → c(c(c(C(x))))
b(B(x)) → C(x)
c(C(x)) → B(x)
b(c(C(x))) → c(c(c(c(C(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
              ↳ QDPToSRSProof
                ↳ QTRS
                  ↳ QTRS Reverse
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                      ↳ QTRS Reverse
                      ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(b(x)) → C1(c(c(x)))
B1(b(x)) → C1(c(c(c(x))))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(b(x)) → C1(c(c(c(x)))) at position [0] we obtained the following new rules:

B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(C(x0))) → C1(c(c(B(x0))))
B1(b(x0)) → C1(c(b(c(a(x0)))))
B1(b(y0)) → C1(b(c(a(c(y0)))))



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

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

B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x)) → C1(c(c(x)))
B1(b(x0)) → C1(c(b(c(a(x0)))))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
B1(b(C(x0))) → C1(c(c(B(x0))))
C1(c(a(x0))) → B1(c(x0))
B1(b(y0)) → C1(b(c(a(c(y0)))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(b(x)) → C1(c(c(x))) at position [0] we obtained the following new rules:

B1(b(x0)) → C1(b(c(a(x0))))
B1(b(c(x0))) → C1(c(b(c(a(x0)))))
B1(b(C(x0))) → C1(c(B(x0)))



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

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

B1(b(x0)) → C1(b(c(a(x0))))
B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x0)) → C1(c(b(c(a(x0)))))
B1(b(x)) → C1(x)
B1(b(x)) → C1(c(x))
B1(b(C(x0))) → C1(c(c(B(x0))))
C1(c(a(x0))) → B1(c(x0))
B1(b(y0)) → C1(b(c(a(c(y0)))))
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(b(c(x0))) → C1(c(b(c(a(x0)))))
C1(c(a(x0))) → C1(x0)
B1(b(C(x0))) → C1(c(B(x0)))

The TRS R consists of the following rules:

a(a(x)) → x
b(b(x)) → c(c(c(c(x))))
c(c(x)) → b(c(a(x)))
b(B(x)) → c(C(x))
b(B(x)) → c(c(C(x)))
b(B(x)) → c(c(c(C(x))))
b(B(x)) → C(x)
c(C(x)) → B(x)
b(c(C(x))) → c(c(c(c(C(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
              ↳ QDPToSRSProof
                ↳ QTRS
                  ↳ QTRS Reverse
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ Narrowing
                                                                            ↳ QDP
                                                                              ↳ DependencyGraphProof
                                                                                ↳ QDP
                                                                                  ↳ Narrowing
                      ↳ QTRS Reverse
                      ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(b(x0)) → C1(b(c(a(x0))))
B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x0)) → C1(c(b(c(a(x0)))))
B1(b(x)) → C1(x)
B1(c(C(x))) → C1(c(c(c(C(x)))))
B1(b(c(x0))) → C1(c(b(c(a(x0)))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
B1(b(C(x0))) → C1(c(c(B(x0))))
C1(c(a(x0))) → B1(c(x0))
B1(b(y0)) → C1(b(c(a(c(y0)))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(c(C(x))) → C1(c(c(c(C(x))))) at position [0] we obtained the following new rules:

B1(c(C(x0))) → C1(c(c(B(x0))))
B1(c(C(y0))) → C1(c(b(c(a(C(y0))))))
B1(c(C(y0))) → C1(b(c(a(c(C(y0))))))



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

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

B1(b(x0)) → C1(b(c(a(x0))))
B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x0)) → C1(c(b(c(a(x0)))))
B1(b(x)) → C1(x)
B1(c(C(x0))) → C1(c(c(B(x0))))
B1(c(C(y0))) → C1(c(b(c(a(C(y0))))))
B1(b(x)) → C1(c(x))
B1(b(C(x0))) → C1(c(c(B(x0))))
C1(c(a(x0))) → B1(c(x0))
B1(b(y0)) → C1(b(c(a(c(y0)))))
B1(c(C(y0))) → C1(b(c(a(c(C(y0))))))
B1(b(c(x0))) → C1(c(b(c(a(x0)))))
C1(c(a(x0))) → C1(x0)

The TRS R consists of the following rules:

a(a(x)) → x
b(b(x)) → c(c(c(c(x))))
c(c(x)) → b(c(a(x)))
b(B(x)) → c(C(x))
b(B(x)) → c(c(C(x)))
b(B(x)) → c(c(c(C(x))))
b(B(x)) → C(x)
c(C(x)) → B(x)
b(c(C(x))) → c(c(c(c(C(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
              ↳ QDPToSRSProof
                ↳ QTRS
                  ↳ QTRS Reverse
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ Narrowing
                                                                            ↳ QDP
                                                                              ↳ DependencyGraphProof
                                                                                ↳ QDP
                                                                                  ↳ Narrowing
                                                                                    ↳ QDP
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                      ↳ QTRS Reverse
                      ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(b(x0)) → C1(b(c(a(x0))))
B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x0)) → C1(c(b(c(a(x0)))))
B1(b(x)) → C1(x)
B1(c(C(y0))) → C1(b(c(a(c(C(y0))))))
B1(c(C(x0))) → C1(c(c(B(x0))))
B1(b(c(x0))) → C1(c(b(c(a(x0)))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
B1(b(C(x0))) → C1(c(c(B(x0))))
C1(c(a(x0))) → B1(c(x0))
B1(b(y0)) → C1(b(c(a(c(y0)))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(b(C(x0))) → C1(c(c(B(x0)))) at position [0] we obtained the following new rules:

B1(b(C(y0))) → C1(b(c(a(B(y0)))))



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

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

B1(b(x0)) → C1(b(c(a(x0))))
B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(C(y0))) → C1(b(c(a(B(y0)))))
B1(b(x)) → C1(x)
B1(b(x0)) → C1(c(b(c(a(x0)))))
B1(c(C(x0))) → C1(c(c(B(x0))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → B1(c(x0))
B1(b(y0)) → C1(b(c(a(c(y0)))))
B1(c(C(y0))) → C1(b(c(a(c(C(y0))))))
B1(b(c(x0))) → C1(c(b(c(a(x0)))))
C1(c(a(x0))) → C1(x0)

The TRS R consists of the following rules:

a(a(x)) → x
b(b(x)) → c(c(c(c(x))))
c(c(x)) → b(c(a(x)))
b(B(x)) → c(C(x))
b(B(x)) → c(c(C(x)))
b(B(x)) → c(c(c(C(x))))
b(B(x)) → C(x)
c(C(x)) → B(x)
b(c(C(x))) → c(c(c(c(C(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
              ↳ QDPToSRSProof
                ↳ QTRS
                  ↳ QTRS Reverse
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ Narrowing
                                                                            ↳ QDP
                                                                              ↳ DependencyGraphProof
                                                                                ↳ QDP
                                                                                  ↳ Narrowing
                                                                                    ↳ QDP
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                      ↳ QTRS Reverse
                      ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(b(x0)) → C1(b(c(a(x0))))
B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x0)) → C1(c(b(c(a(x0)))))
B1(b(x)) → C1(x)
B1(c(C(y0))) → C1(b(c(a(c(C(y0))))))
B1(c(C(x0))) → C1(c(c(B(x0))))
B1(b(c(x0))) → C1(c(b(c(a(x0)))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))
B1(b(y0)) → C1(b(c(a(c(y0)))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(b(x0)) → C1(c(b(c(a(x0))))) at position [0] we obtained the following new rules:

B1(b(a(x0))) → C1(c(b(c(x0))))



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

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

B1(b(x0)) → C1(b(c(a(x0))))
B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x)) → C1(x)
B1(c(C(y0))) → C1(b(c(a(c(C(y0))))))
B1(c(C(x0))) → C1(c(c(B(x0))))
B1(b(c(x0))) → C1(c(b(c(a(x0)))))
B1(b(a(x0))) → C1(c(b(c(x0))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))
B1(b(y0)) → C1(b(c(a(c(y0)))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(b(x0)) → C1(b(c(a(x0)))) at position [0] we obtained the following new rules:

B1(b(a(x0))) → C1(b(c(x0)))



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

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

B1(b(a(x0))) → C1(b(c(x0)))
B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x)) → C1(x)
B1(c(C(y0))) → C1(b(c(a(c(C(y0))))))
B1(c(C(x0))) → C1(c(c(B(x0))))
B1(b(c(x0))) → C1(c(b(c(a(x0)))))
B1(b(x)) → C1(c(x))
B1(b(a(x0))) → C1(c(b(c(x0))))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))
B1(b(y0)) → C1(b(c(a(c(y0)))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(b(c(x0))) → C1(c(b(c(a(x0))))) at position [0] we obtained the following new rules:

B1(b(c(a(x0)))) → C1(c(b(c(x0))))



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

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

B1(b(a(x0))) → C1(b(c(x0)))
B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x)) → C1(x)
B1(c(C(y0))) → C1(b(c(a(c(C(y0))))))
B1(c(C(x0))) → C1(c(c(B(x0))))
B1(b(a(x0))) → C1(c(b(c(x0))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
B1(b(c(a(x0)))) → C1(c(b(c(x0))))
C1(c(a(x0))) → B1(c(x0))
B1(b(y0)) → C1(b(c(a(c(y0)))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(c(C(x0))) → C1(c(c(B(x0)))) at position [0] we obtained the following new rules:

B1(c(C(y0))) → C1(b(c(a(B(y0)))))



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

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

B1(c(C(y0))) → C1(b(c(a(B(y0)))))
B1(b(a(x0))) → C1(b(c(x0)))
B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x)) → C1(x)
B1(c(C(y0))) → C1(b(c(a(c(C(y0))))))
B1(b(x)) → C1(c(x))
B1(b(a(x0))) → C1(c(b(c(x0))))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))
B1(b(c(a(x0)))) → C1(c(b(c(x0))))
B1(b(y0)) → C1(b(c(a(c(y0)))))

The TRS R consists of the following rules:

a(a(x)) → x
b(b(x)) → c(c(c(c(x))))
c(c(x)) → b(c(a(x)))
b(B(x)) → c(C(x))
b(B(x)) → c(c(C(x)))
b(B(x)) → c(c(c(C(x))))
b(B(x)) → C(x)
c(C(x)) → B(x)
b(c(C(x))) → c(c(c(c(C(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
              ↳ QDPToSRSProof
                ↳ QTRS
                  ↳ QTRS Reverse
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ Narrowing
                                                                            ↳ QDP
                                                                              ↳ DependencyGraphProof
                                                                                ↳ QDP
                                                                                  ↳ Narrowing
                                                                                    ↳ QDP
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ Narrowing
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                      ↳ QTRS Reverse
                      ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(b(a(x0))) → C1(b(c(x0)))
B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x)) → C1(x)
B1(c(C(y0))) → C1(b(c(a(c(C(y0))))))
B1(b(a(x0))) → C1(c(b(c(x0))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
B1(b(c(a(x0)))) → C1(c(b(c(x0))))
C1(c(a(x0))) → B1(c(x0))
B1(b(y0)) → C1(b(c(a(c(y0)))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(c(C(y0))) → C1(b(c(a(c(C(y0)))))) at position [0] we obtained the following new rules:

B1(c(C(x0))) → C1(b(c(a(B(x0)))))



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

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

B1(c(C(x0))) → C1(b(c(a(B(x0)))))
B1(b(a(x0))) → C1(b(c(x0)))
B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x)) → C1(x)
B1(b(x)) → C1(c(x))
B1(b(a(x0))) → C1(c(b(c(x0))))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))
B1(b(c(a(x0)))) → C1(c(b(c(x0))))
B1(b(y0)) → C1(b(c(a(c(y0)))))

The TRS R consists of the following rules:

a(a(x)) → x
b(b(x)) → c(c(c(c(x))))
c(c(x)) → b(c(a(x)))
b(B(x)) → c(C(x))
b(B(x)) → c(c(C(x)))
b(B(x)) → c(c(c(C(x))))
b(B(x)) → C(x)
c(C(x)) → B(x)
b(c(C(x))) → c(c(c(c(C(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
              ↳ QDPToSRSProof
                ↳ QTRS
                  ↳ QTRS Reverse
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ Narrowing
                                                                            ↳ QDP
                                                                              ↳ DependencyGraphProof
                                                                                ↳ QDP
                                                                                  ↳ Narrowing
                                                                                    ↳ QDP
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ Narrowing
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ QDPOrderProof
                      ↳ QTRS Reverse
                      ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(b(a(x0))) → C1(b(c(x0)))
B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x)) → C1(x)
B1(b(a(x0))) → C1(c(b(c(x0))))
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
B1(b(c(a(x0)))) → C1(c(b(c(x0))))
C1(c(a(x0))) → B1(c(x0))
B1(b(y0)) → C1(b(c(a(c(y0)))))

The TRS R consists of the following rules:

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


B1(b(a(x0))) → C1(b(c(x0)))
B1(b(a(x0))) → C1(c(b(c(x0))))
The remaining pairs can at least be oriented weakly.

B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x)) → C1(x)
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
B1(b(c(a(x0)))) → C1(c(b(c(x0))))
C1(c(a(x0))) → B1(c(x0))
B1(b(y0)) → C1(b(c(a(c(y0)))))
Used ordering: Polynomial Order [21,25] with Interpretation:

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


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


POL( B(x1) ) = 0


POL( a(x1) ) = x1 + 1


POL( B1(x1) ) = x1 + 1


POL( b(x1) ) = x1


POL( C1(x1) ) = 1



The following usable rules [17] were oriented:

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



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

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

B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x)) → C1(x)
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
C1(c(a(x0))) → B1(c(x0))
B1(b(c(a(x0)))) → C1(c(b(c(x0))))
B1(b(y0)) → C1(b(c(a(c(y0)))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We found the following model for the rules of the TRS R. Interpretation over the domain with elements from 0 to 1.C: 0
c: 0
B: 0
a: 1 + x0
B1: 0
b: 0
C1: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:

B1.0(b.0(c.0(a.1(x0)))) → C1.0(c.0(b.0(c.1(x0))))
B1.0(b.0(c.0(x0))) → C1.0(c.0(c.0(b.0(c.1(a.0(x0))))))
B1.0(b.1(y0)) → C1.0(b.0(c.1(a.0(c.1(y0)))))
C1.0(c.0(a.1(x0))) → B1.0(c.1(x0))
B1.0(b.0(c.1(a.0(x0)))) → C1.0(c.0(b.0(c.0(x0))))
C1.0(c.1(a.0(x0))) → B1.0(c.0(x0))
C1.0(c.0(a.1(x0))) → C1.1(x0)
B1.0(b.0(x)) → C1.0(c.0(x))
B1.0(b.0(c.1(x0))) → C1.0(c.0(c.0(b.0(c.0(a.1(x0))))))
C1.0(c.1(a.0(x0))) → C1.0(x0)
B1.0(b.1(x)) → C1.1(x)
B1.0(b.1(x)) → C1.0(c.1(x))
B1.0(b.0(y0)) → C1.0(b.0(c.1(a.0(c.0(y0)))))
B1.0(b.0(x)) → C1.0(x)

The TRS R consists of the following rules:

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

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

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

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

B1.0(b.0(c.0(a.1(x0)))) → C1.0(c.0(b.0(c.1(x0))))
B1.0(b.0(c.0(x0))) → C1.0(c.0(c.0(b.0(c.1(a.0(x0))))))
B1.0(b.1(y0)) → C1.0(b.0(c.1(a.0(c.1(y0)))))
C1.0(c.0(a.1(x0))) → B1.0(c.1(x0))
B1.0(b.0(c.1(a.0(x0)))) → C1.0(c.0(b.0(c.0(x0))))
C1.0(c.1(a.0(x0))) → B1.0(c.0(x0))
C1.0(c.0(a.1(x0))) → C1.1(x0)
B1.0(b.0(x)) → C1.0(c.0(x))
B1.0(b.0(c.1(x0))) → C1.0(c.0(c.0(b.0(c.0(a.1(x0))))))
C1.0(c.1(a.0(x0))) → C1.0(x0)
B1.0(b.1(x)) → C1.1(x)
B1.0(b.1(x)) → C1.0(c.1(x))
B1.0(b.0(y0)) → C1.0(b.0(c.1(a.0(c.0(y0)))))
B1.0(b.0(x)) → C1.0(x)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 6 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ QDPToSRSProof
                ↳ QTRS
                  ↳ QTRS Reverse
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ Narrowing
                                                                            ↳ QDP
                                                                              ↳ DependencyGraphProof
                                                                                ↳ QDP
                                                                                  ↳ Narrowing
                                                                                    ↳ QDP
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ Narrowing
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ QDPOrderProof
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ SemLabProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ DependencyGraphProof
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ UsableRulesReductionPairsProof
                                                                                                                                  ↳ SemLabProof2
                      ↳ QTRS Reverse
                      ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C1.0(c.1(a.0(x0))) → B1.0(c.0(x0))
B1.0(b.0(x)) → C1.0(c.0(x))
B1.0(b.0(c.1(x0))) → C1.0(c.0(c.0(b.0(c.0(a.1(x0))))))
C1.0(c.1(a.0(x0))) → C1.0(x0)
B1.0(b.1(x)) → C1.0(c.1(x))
B1.0(b.0(c.0(x0))) → C1.0(c.0(c.0(b.0(c.1(a.0(x0))))))
B1.0(b.0(c.1(a.0(x0)))) → C1.0(c.0(b.0(c.0(x0))))
B1.0(b.0(x)) → C1.0(x)

The TRS R consists of the following rules:

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

B1.0(b.1(x)) → C1.0(c.1(x))
The following rules are removed from R:

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

POL(B.0(x1)) = x1   
POL(B.1(x1)) = x1   
POL(B1.0(x1)) = x1   
POL(C.0(x1)) = x1   
POL(C.1(x1)) = x1   
POL(C1.0(x1)) = x1   
POL(a.0(x1)) = x1   
POL(a.1(x1)) = x1   
POL(b.0(x1)) = x1   
POL(b.1(x1)) = 1 + x1   
POL(c.0(x1)) = x1   
POL(c.1(x1)) = x1   



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ QDPToSRSProof
                ↳ QTRS
                  ↳ QTRS Reverse
                    ↳ QTRS
                      ↳ DependencyPairsProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ Narrowing
                                                                            ↳ QDP
                                                                              ↳ DependencyGraphProof
                                                                                ↳ QDP
                                                                                  ↳ Narrowing
                                                                                    ↳ QDP
                                                                                      ↳ DependencyGraphProof
                                                                                        ↳ QDP
                                                                                          ↳ Narrowing
                                                                                            ↳ QDP
                                                                                              ↳ DependencyGraphProof
                                                                                                ↳ QDP
                                                                                                  ↳ Narrowing
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ Narrowing
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ QDPOrderProof
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ SemLabProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ DependencyGraphProof
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ UsableRulesReductionPairsProof
                                                                                                                                            ↳ QDP
                                                                                                                                  ↳ SemLabProof2
                      ↳ QTRS Reverse
                      ↳ QTRS Reverse
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

C1.0(c.1(a.0(x0))) → B1.0(c.0(x0))
B1.0(b.0(x)) → C1.0(c.0(x))
B1.0(b.0(c.1(x0))) → C1.0(c.0(c.0(b.0(c.0(a.1(x0))))))
C1.0(c.1(a.0(x0))) → C1.0(x0)
B1.0(b.0(c.0(x0))) → C1.0(c.0(c.0(b.0(c.1(a.0(x0))))))
B1.0(b.0(c.1(a.0(x0)))) → C1.0(c.0(b.0(c.0(x0))))
B1.0(b.0(x)) → C1.0(x)

The TRS R consists of the following rules:

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

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

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

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

B1(b(c(x0))) → C1(c(c(b(c(a(x0))))))
B1(b(x)) → C1(x)
B1(b(x)) → C1(c(x))
C1(c(a(x0))) → C1(x0)
B1(b(c(a(x0)))) → C1(c(b(c(x0))))
C1(c(a(x0))) → B1(c(x0))

The TRS R consists of the following rules:

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

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

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

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

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

The set Q is empty.

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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

↳ QTRS
  ↳ DependencyPairsProof
  ↳ QTRS Reverse
    ↳ QTRS
  ↳ QTRS Reverse

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

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

Q is empty.

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

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

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

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

The set Q is empty.

↳ QTRS
  ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse
    ↳ QTRS

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

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

Q is empty.