Termination w.r.t. Q proof of /tmp/tmp76R7Mk/secr2.srs

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

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

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

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

Q is empty.

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

A(b(c(y0))) → B(c(c(y0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

A(c(c(y0))) → B(c(c(y0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

A(b(c(y0))) → B(c(y0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

A(c(c(y0))) → B(c(y0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

A(b(c(x0))) → B(x0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

A(c(c(x0))) → B(x0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

Q is empty.

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

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

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

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

The set Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

Q is empty.

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

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

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

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

The set Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                                ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

Q is empty.

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

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

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

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

The set Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                                ↳ QTRS

                                                              ↳ DependencyPairsProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

Q is empty.

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 14 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                        ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                      ↳ QDP

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesReductionPairsProof

                                                                        ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                      ↳ QDP

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

The following dependency pairs can be deleted:

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

The following rules are removed from R:

a(c(A(x))) → a(a(a(B(x))))

Used ordering: POLO with Polynomial interpretation [25]:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesReductionPairsProof

                                                                              ↳ QDP

                                                                                ↳ RuleRemovalProof

                                                                        ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                      ↳ QDP

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

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

Strictly oriented rules of the TRS R:

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

Used ordering: POLO with Polynomial interpretation [25]:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesReductionPairsProof

                                                                              ↳ QDP

                                                                                ↳ RuleRemovalProof

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                        ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                      ↳ QDP

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesReductionPairsProof

                                                                              ↳ QDP

                                                                                ↳ RuleRemovalProof

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ RuleRemovalProof

                                                                        ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                      ↳ QDP

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

A¹(b(A(x))) → A¹(a(b(B(x))))

Strictly oriented rules of the TRS R:

a(x) → x

Used ordering: POLO with Polynomial interpretation [25]:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesReductionPairsProof

                                                                              ↳ QDP

                                                                                ↳ RuleRemovalProof

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ RuleRemovalProof

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                        ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                      ↳ QDP

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

A¹(b(A(x))) → A¹(a(a(b(B(x)))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                        ↳ UsableRulesProof

                                                                      ↳ QDP

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                        ↳ UsableRulesProof

                                                                      ↳ QDP

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

Strictly oriented rules of the TRS R:

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

Used ordering: POLO with Polynomial interpretation [25]:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                        ↳ UsableRulesProof

                                                                      ↳ QDP

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                      ↳ QDP

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

C(a(x)) → C(x)

Strictly oriented rules of the TRS R:

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

Used ordering: POLO with Polynomial interpretation [25]:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial Order [21,25] with Interpretation:

POL( A(x₁) ) = x₁ + 1

POL( C(x₁) ) = x₁

POL( c(x₁) ) = x₁

POL( b(x₁) ) = x₁

POL( B(x₁) ) = max{0, x₁ - 1}

POL( a(x₁) ) = max{0, -1}

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

C(b(A(y0))) → C(B(y0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

C(b(b(A(y0)))) → C(B(y0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

C(b(c(A(y0)))) → C(B(y0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial Order [21,25] with Interpretation:

POL( A(x₁) ) = 1

POL( C(x₁) ) = x₁

POL( c(x₁) ) = x₁

POL( b(x₁) ) = x₁

POL( B(x₁) ) = max{0, -1}

POL( a(x₁) ) = max{0, -1}

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ SemLabProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ SemLabProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ DependencyGraphProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 10 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ SemLabProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ UsableRulesReductionPairsProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

The following rules are removed from R:

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

Used ordering: POLO with Polynomial interpretation [25]:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ SemLabProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ UsableRulesReductionPairsProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesReductionPairsProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

The following rules are removed from R:

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

Used ordering: POLO with Polynomial interpretation [25]:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ RuleRemovalProof

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ SemLabProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ DependencyGraphProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ UsableRulesReductionPairsProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ UsableRulesReductionPairsProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ PisEmptyProof

                                                                        ↳ UsableRulesProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ Narrowing

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ Narrowing

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ QDPToSRSProof

                                                        ↳ QTRS

                                                          ↳ QTRS Reverse

                                                            ↳ QTRS

                                                              ↳ QTRS Reverse

                                                              ↳ QTRS Reverse

                                                              ↳ DependencyPairsProof

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ AND

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

  ↳ QTRS Reverse

  ↳ QTRS Reverse

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The set Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

  ↳ QTRS Reverse

    ↳ QTRS

  ↳ QTRS Reverse

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

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

Q is empty.

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

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

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

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

The set Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

  ↳ QTRS Reverse

  ↳ QTRS Reverse

    ↳ QTRS

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

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

Q is empty.