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

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

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

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

Q is empty.

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ 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:

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

The TRS R consists of the following rules:

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

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ 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(a(x0)) → B(b(a(x0)))
B(b(c(x0))) → A(c(c(x0)))
C(c(a(x0))) → B(b(a(b(x0))))
B(b(b(x0))) → C(a(c(b(x0))))
A(b(x1)) → A(x1)
A(a(x1)) → B(x1)
A(a(x1)) → A(b(x1))
A(b(x1)) → B(a(x1))
C(c(x1)) → A(x1)
C(c(b(x0))) → B(b(a(x0)))
A(a(c(x0))) → B(a(c(x0)))
A(a(b(x0))) → B(a(a(c(b(x0)))))
C(c(a(x0))) → C(b(b(a(b(x0)))))
B(b(c(x0))) → C(c(x0))
B(b(b(x0))) → A(c(a(c(b(x0)))))
C(c(b(x0))) → C(b(b(a(x0))))

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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.


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

B(b(c(x0))) → A(c(c(x0)))
A(a(x0)) → B(b(a(x0)))
C(c(a(x0))) → B(b(a(b(x0))))
A(b(b(x0))) → B(b(a(x0)))
A(a(x1)) → B(x1)
A(b(x1)) → A(x1)
A(a(x1)) → A(b(x1))
A(b(a(x0))) → B(b(a(b(x0))))
C(c(x1)) → A(x1)
A(a(c(x0))) → B(a(c(x0)))
C(c(b(x0))) → B(b(a(x0)))
A(a(b(x0))) → B(a(a(c(b(x0)))))
B(b(b(x0))) → A(c(a(c(b(x0)))))
B(b(c(x0))) → C(c(x0))
Used ordering: Polynomial Order [21,25] with Interpretation:

POL( A(x1) ) = 1


POL( C(x1) ) = x1


POL( c(x1) ) = 1


POL( b(x1) ) = x1


POL( B(x1) ) = 1


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



The following usable rules [17] were oriented:

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



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

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

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

The TRS R consists of the following rules:

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

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: 0
a: 0
A: 0
b: x0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:

A.0(a.1(x1)) → A.1(b.1(x1))
B.1(b.1(b.1(x0))) → A.1(c.0(a.1(c.1(b.1(x0)))))
A.0(a.1(c.1(x0))) → B.0(a.1(c.1(x0)))
B.0(b.0(b.0(x0))) → A.1(c.0(a.1(c.0(b.0(x0)))))
B.1(b.1(c.1(x0))) → C.1(c.1(x0))
C.1(c.1(x1)) → A.0(x1)
B.1(b.1(c.0(x0))) → C.0(c.0(x0))
C.1(c.0(a.0(x0))) → B.0(b.0(a.0(b.0(x0))))
A.1(b.1(b.1(x0))) → B.0(b.0(a.1(x0)))
A.1(b.1(x1)) → A.1(x1)
A.0(a.1(x0)) → B.0(b.0(a.1(x0)))
C.1(c.1(b.1(x0))) → B.0(b.0(a.1(x0)))
C.1(c.0(a.1(x0))) → B.0(b.0(a.1(b.1(x0))))
B.1(b.1(b.1(x0))) → A.0(c.0(a.1(c.1(b.1(x0)))))
C.1(c.0(x1)) → A.0(x1)
B.0(b.0(b.0(x0))) → A.0(c.0(a.1(c.0(b.0(x0)))))
A.0(b.0(x1)) → A.0(x1)
B.1(b.1(c.0(x0))) → A.0(c.1(c.0(x0)))
A.0(a.0(x1)) → B.0(x1)
B.1(b.1(c.0(x0))) → A.1(c.1(c.0(x0)))
A.0(a.0(x1)) → A.0(b.0(x1))
A.0(b.0(a.1(x0))) → B.0(b.0(a.1(b.1(x0))))
A.0(a.0(b.0(x0))) → B.0(a.0(a.1(c.0(b.0(x0)))))
C.1(c.1(x1)) → A.1(x1)
A.0(a.1(x1)) → B.0(x1)
A.0(a.1(c.0(x0))) → B.0(a.1(c.0(x0)))
A.0(a.1(x1)) → B.1(x1)
A.0(b.0(a.0(x0))) → B.0(b.0(a.0(b.0(x0))))
C.1(c.0(b.0(x0))) → B.0(b.0(a.0(x0)))
A.0(a.1(x1)) → A.0(b.1(x1))
B.1(b.1(c.1(x0))) → A.0(c.1(c.1(x0)))
B.1(b.1(c.1(x0))) → C.0(c.1(x0))
A.0(a.0(x0)) → B.0(b.0(a.0(x0)))
B.1(b.1(c.0(x0))) → C.1(c.0(x0))
B.1(b.1(c.1(x0))) → A.1(c.1(c.1(x0)))
A.0(a.1(b.1(x0))) → B.0(a.0(a.1(c.1(b.1(x0)))))
A.0(b.0(b.0(x0))) → B.0(b.0(a.0(x0)))
A.1(b.1(x1)) → A.0(x1)

The TRS R consists of the following rules:

a.0(a.0(x1)) → b.0(a.0(b.0(x1)))
a.0(b.0(x1)) → b.0(a.0(x1))
c.1(x0) → c.0(x0)
b.1(c.0(x1)) → c.0(x1)
b.1(x0) → b.0(x0)
b.1(c.1(x1)) → c.1(x1)
a.1(x0) → a.0(x0)
a.1(b.1(x1)) → b.0(a.1(x1))
c.1(c.0(x1)) → c.0(b.0(a.0(x1)))
a.0(a.1(x1)) → b.0(a.1(b.1(x1)))
b.1(b.1(x1)) → a.1(c.1(b.1(x1)))
c.1(c.1(x1)) → c.0(b.0(a.1(x1)))
b.0(b.0(x1)) → a.1(c.0(b.0(x1)))

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

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

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

A.0(a.1(x1)) → A.1(b.1(x1))
B.1(b.1(b.1(x0))) → A.1(c.0(a.1(c.1(b.1(x0)))))
A.0(a.1(c.1(x0))) → B.0(a.1(c.1(x0)))
B.0(b.0(b.0(x0))) → A.1(c.0(a.1(c.0(b.0(x0)))))
B.1(b.1(c.1(x0))) → C.1(c.1(x0))
C.1(c.1(x1)) → A.0(x1)
B.1(b.1(c.0(x0))) → C.0(c.0(x0))
C.1(c.0(a.0(x0))) → B.0(b.0(a.0(b.0(x0))))
A.1(b.1(b.1(x0))) → B.0(b.0(a.1(x0)))
A.1(b.1(x1)) → A.1(x1)
A.0(a.1(x0)) → B.0(b.0(a.1(x0)))
C.1(c.1(b.1(x0))) → B.0(b.0(a.1(x0)))
C.1(c.0(a.1(x0))) → B.0(b.0(a.1(b.1(x0))))
B.1(b.1(b.1(x0))) → A.0(c.0(a.1(c.1(b.1(x0)))))
C.1(c.0(x1)) → A.0(x1)
B.0(b.0(b.0(x0))) → A.0(c.0(a.1(c.0(b.0(x0)))))
A.0(b.0(x1)) → A.0(x1)
B.1(b.1(c.0(x0))) → A.0(c.1(c.0(x0)))
A.0(a.0(x1)) → B.0(x1)
B.1(b.1(c.0(x0))) → A.1(c.1(c.0(x0)))
A.0(a.0(x1)) → A.0(b.0(x1))
A.0(b.0(a.1(x0))) → B.0(b.0(a.1(b.1(x0))))
A.0(a.0(b.0(x0))) → B.0(a.0(a.1(c.0(b.0(x0)))))
C.1(c.1(x1)) → A.1(x1)
A.0(a.1(x1)) → B.0(x1)
A.0(a.1(c.0(x0))) → B.0(a.1(c.0(x0)))
A.0(a.1(x1)) → B.1(x1)
A.0(b.0(a.0(x0))) → B.0(b.0(a.0(b.0(x0))))
C.1(c.0(b.0(x0))) → B.0(b.0(a.0(x0)))
A.0(a.1(x1)) → A.0(b.1(x1))
B.1(b.1(c.1(x0))) → A.0(c.1(c.1(x0)))
B.1(b.1(c.1(x0))) → C.0(c.1(x0))
A.0(a.0(x0)) → B.0(b.0(a.0(x0)))
B.1(b.1(c.0(x0))) → C.1(c.0(x0))
B.1(b.1(c.1(x0))) → A.1(c.1(c.1(x0)))
A.0(a.1(b.1(x0))) → B.0(a.0(a.1(c.1(b.1(x0)))))
A.0(b.0(b.0(x0))) → B.0(b.0(a.0(x0)))
A.1(b.1(x1)) → A.0(x1)

The TRS R consists of the following rules:

a.0(a.0(x1)) → b.0(a.0(b.0(x1)))
a.0(b.0(x1)) → b.0(a.0(x1))
c.1(x0) → c.0(x0)
b.1(c.0(x1)) → c.0(x1)
b.1(x0) → b.0(x0)
b.1(c.1(x1)) → c.1(x1)
a.1(x0) → a.0(x0)
a.1(b.1(x1)) → b.0(a.1(x1))
c.1(c.0(x1)) → c.0(b.0(a.0(x1)))
a.0(a.1(x1)) → b.0(a.1(b.1(x1)))
b.1(b.1(x1)) → a.1(c.1(b.1(x1)))
c.1(c.1(x1)) → c.0(b.0(a.1(x1)))
b.0(b.0(x1)) → a.1(c.0(b.0(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 24 less nodes.

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

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

A.0(a.1(x1)) → A.1(b.1(x1))
C.1(c.0(x1)) → A.0(x1)
A.0(b.0(x1)) → A.0(x1)
B.1(b.1(c.0(x0))) → A.0(c.1(c.0(x0)))
A.0(a.0(x1)) → A.0(b.0(x1))
B.1(b.1(c.1(x0))) → C.1(c.1(x0))
C.1(c.1(x1)) → A.1(x1)
C.1(c.1(x1)) → A.0(x1)
A.0(a.1(x1)) → B.1(x1)
A.1(b.1(x1)) → A.1(x1)
A.0(a.1(x1)) → A.0(b.1(x1))
B.1(b.1(c.1(x0))) → A.0(c.1(c.1(x0)))
B.1(b.1(c.0(x0))) → C.1(c.0(x0))
A.1(b.1(x1)) → A.0(x1)

The TRS R consists of the following rules:

a.0(a.0(x1)) → b.0(a.0(b.0(x1)))
a.0(b.0(x1)) → b.0(a.0(x1))
c.1(x0) → c.0(x0)
b.1(c.0(x1)) → c.0(x1)
b.1(x0) → b.0(x0)
b.1(c.1(x1)) → c.1(x1)
a.1(x0) → a.0(x0)
a.1(b.1(x1)) → b.0(a.1(x1))
c.1(c.0(x1)) → c.0(b.0(a.0(x1)))
a.0(a.1(x1)) → b.0(a.1(b.1(x1)))
b.1(b.1(x1)) → a.1(c.1(b.1(x1)))
c.1(c.1(x1)) → c.0(b.0(a.1(x1)))
b.0(b.0(x1)) → a.1(c.0(b.0(x1)))

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

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

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

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

The TRS R consists of the following rules:

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

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: 0
B: 0
a: 1
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:

A.1(a.0(x1)) → A.1(b.0(x1))
B.1(b.0(c.1(x0))) → C.0(c.1(x0))
B.1(b.0(c.0(x0))) → C.0(c.0(x0))
A.1(a.1(x1)) → B.1(x1)
A.1(b.0(x1)) → A.0(x1)
A.1(a.1(x1)) → A.1(b.1(x1))
A.1(a.0(x1)) → A.0(b.0(x1))
A.1(a.0(x1)) → B.0(x1)
A.1(a.1(x1)) → B.0(x1)
A.1(a.1(x1)) → A.0(b.1(x1))
C.0(c.0(x1)) → A.0(x1)
C.0(c.1(x1)) → A.0(x1)
A.1(b.1(x1)) → A.1(x1)
B.1(b.0(c.1(x0))) → A.0(c.0(c.1(x0)))
B.1(b.0(c.0(x0))) → A.0(c.0(c.0(x0)))
C.0(c.1(x1)) → A.1(x1)
A.1(b.1(x1)) → A.0(x1)

The TRS R consists of the following rules:

b.1(b.0(x1)) → a.0(c.1(b.0(x1)))
c.1(x0) → c.0(x0)
b.1(b.1(x1)) → a.0(c.1(b.1(x1)))
b.1(x0) → b.0(x0)
b.0(c.1(x1)) → c.1(x1)
a.1(a.1(x1)) → b.1(a.1(b.1(x1)))
c.0(c.1(x1)) → c.1(b.1(a.1(x1)))
a.1(x0) → a.0(x0)
a.1(b.0(x1)) → b.1(a.0(x1))
b.0(c.0(x1)) → c.0(x1)
a.1(a.0(x1)) → b.1(a.1(b.0(x1)))
c.0(c.0(x1)) → c.1(b.1(a.0(x1)))
a.1(b.1(x1)) → b.1(a.1(x1))

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

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

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

A.1(a.0(x1)) → A.1(b.0(x1))
B.1(b.0(c.1(x0))) → C.0(c.1(x0))
B.1(b.0(c.0(x0))) → C.0(c.0(x0))
A.1(a.1(x1)) → B.1(x1)
A.1(b.0(x1)) → A.0(x1)
A.1(a.1(x1)) → A.1(b.1(x1))
A.1(a.0(x1)) → A.0(b.0(x1))
A.1(a.0(x1)) → B.0(x1)
A.1(a.1(x1)) → B.0(x1)
A.1(a.1(x1)) → A.0(b.1(x1))
C.0(c.0(x1)) → A.0(x1)
C.0(c.1(x1)) → A.0(x1)
A.1(b.1(x1)) → A.1(x1)
B.1(b.0(c.1(x0))) → A.0(c.0(c.1(x0)))
B.1(b.0(c.0(x0))) → A.0(c.0(c.0(x0)))
C.0(c.1(x1)) → A.1(x1)
A.1(b.1(x1)) → A.0(x1)

The TRS R consists of the following rules:

b.1(b.0(x1)) → a.0(c.1(b.0(x1)))
c.1(x0) → c.0(x0)
b.1(b.1(x1)) → a.0(c.1(b.1(x1)))
b.1(x0) → b.0(x0)
b.0(c.1(x1)) → c.1(x1)
a.1(a.1(x1)) → b.1(a.1(b.1(x1)))
c.0(c.1(x1)) → c.1(b.1(a.1(x1)))
a.1(x0) → a.0(x0)
a.1(b.0(x1)) → b.1(a.0(x1))
b.0(c.0(x1)) → c.0(x1)
a.1(a.0(x1)) → b.1(a.1(b.0(x1)))
c.0(c.0(x1)) → c.1(b.1(a.0(x1)))
a.1(b.1(x1)) → b.1(a.1(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 10 less nodes.

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

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

A.1(a.0(x1)) → A.1(b.0(x1))
B.1(b.0(c.1(x0))) → C.0(c.1(x0))
A.1(b.1(x1)) → A.1(x1)
B.1(b.0(c.0(x0))) → C.0(c.0(x0))
C.0(c.1(x1)) → A.1(x1)
A.1(a.1(x1)) → B.1(x1)
A.1(a.1(x1)) → A.1(b.1(x1))

The TRS R consists of the following rules:

b.1(b.0(x1)) → a.0(c.1(b.0(x1)))
c.1(x0) → c.0(x0)
b.1(b.1(x1)) → a.0(c.1(b.1(x1)))
b.1(x0) → b.0(x0)
b.0(c.1(x1)) → c.1(x1)
a.1(a.1(x1)) → b.1(a.1(b.1(x1)))
c.0(c.1(x1)) → c.1(b.1(a.1(x1)))
a.1(x0) → a.0(x0)
a.1(b.0(x1)) → b.1(a.0(x1))
b.0(c.0(x1)) → c.0(x1)
a.1(a.0(x1)) → b.1(a.1(b.0(x1)))
c.0(c.0(x1)) → c.1(b.1(a.0(x1)))
a.1(b.1(x1)) → b.1(a.1(x1))

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

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

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

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

The TRS R consists of the following rules:

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

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


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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

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

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

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

Q is empty.

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

B1(a(x)) → B1(x)
B1(b(x)) → C1(a(x))
B1(a(x)) → A1(b(x))
C1(b(B(x))) → C1(C(x))
C1(c(x)) → A1(b(c(x)))
C1(c(x)) → B1(c(x))
A1(a(x)) → A1(b(x))
A1(a(x)) → B1(x)
C1(b(x)) → C1(x)
B1(b(x)) → A1(x)
B1(b(x)) → B1(c(a(x)))
A1(A(x)) → B1(A(x))
A1(a(x)) → B1(a(b(x)))

The TRS R consists of the following rules:

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

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

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

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

B1(a(x)) → B1(x)
B1(b(x)) → C1(a(x))
B1(a(x)) → A1(b(x))
C1(b(B(x))) → C1(C(x))
C1(c(x)) → A1(b(c(x)))
C1(c(x)) → B1(c(x))
A1(a(x)) → A1(b(x))
A1(a(x)) → B1(x)
C1(b(x)) → C1(x)
B1(b(x)) → A1(x)
B1(b(x)) → B1(c(a(x)))
A1(A(x)) → B1(A(x))
A1(a(x)) → B1(a(b(x)))

The TRS R consists of the following rules:

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

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

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

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

A1(a(x)) → A1(b(x))
A1(a(x)) → B1(x)
C1(b(x)) → C1(x)
B1(a(x)) → B1(x)
B1(b(x)) → A1(x)
B1(a(x)) → A1(b(x))
B1(b(x)) → C1(a(x))
B1(b(x)) → B1(c(a(x)))
C1(c(x)) → B1(c(x))
C1(c(x)) → A1(b(c(x)))
A1(a(x)) → B1(a(b(x)))

The TRS R consists of the following rules:

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

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

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



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

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

B1(a(x)) → B1(x)
B1(a(x)) → A1(b(x))
B1(b(x)) → C1(a(x))
B1(b(A(x0))) → B1(c(b(A(x0))))
C1(c(x)) → B1(c(x))
C1(c(x)) → A1(b(c(x)))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → A1(b(x))
A1(a(x)) → B1(x)
C1(b(x)) → C1(x)
B1(b(A(x0))) → B1(c(B(x0)))
B1(b(x)) → A1(x)
A1(a(x)) → B1(a(b(x)))

The TRS R consists of the following rules:

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

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

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

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

A1(a(x)) → A1(b(x))
A1(a(x)) → B1(x)
C1(b(x)) → C1(x)
B1(a(x)) → B1(x)
B1(b(x)) → A1(x)
B1(a(x)) → A1(b(x))
B1(b(x)) → C1(a(x))
B1(b(A(x0))) → B1(c(b(A(x0))))
C1(c(x)) → B1(c(x))
C1(c(x)) → A1(b(c(x)))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → B1(a(b(x)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(x)) → A1(b(c(x))) at position [0] we obtained the following new rules:

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
C1(c(C(x0))) → A1(b(A(x0)))
C1(c(c(x0))) → A1(b(a(b(c(x0)))))



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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(C(x0))) → A1(b(A(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
B1(a(x)) → B1(x)
B1(b(x)) → C1(a(x))
B1(a(x)) → A1(b(x))
B1(b(A(x0))) → B1(c(b(A(x0))))
C1(c(x)) → B1(c(x))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → A1(b(x))
A1(a(x)) → B1(x)
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
A1(a(x)) → B1(a(b(x)))

The TRS R consists of the following rules:

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

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

B1(b(A(x0))) → C1(B(x0))
B1(b(A(x0))) → C1(b(A(x0)))
B1(b(a(x0))) → C1(b(a(b(x0))))



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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
C1(c(C(x0))) → A1(b(A(x0)))
B1(a(x)) → B1(x)
B1(b(A(x0))) → C1(b(A(x0)))
B1(a(x)) → A1(b(x))
B1(b(A(x0))) → B1(c(b(A(x0))))
B1(b(A(x0))) → C1(B(x0))
C1(c(x)) → B1(c(x))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → A1(b(x))
A1(a(x)) → B1(x)
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
B1(b(a(x0))) → C1(b(a(b(x0))))
A1(a(x)) → B1(a(b(x)))

The TRS R consists of the following rules:

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

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

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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(C(x0))) → A1(b(A(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
B1(a(x)) → B1(x)
B1(b(A(x0))) → C1(b(A(x0)))
B1(a(x)) → A1(b(x))
B1(b(A(x0))) → B1(c(b(A(x0))))
C1(c(x)) → B1(c(x))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → A1(b(x))
A1(a(x)) → B1(x)
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
A1(a(x)) → B1(a(b(x)))
B1(b(a(x0))) → C1(b(a(b(x0))))

The TRS R consists of the following rules:

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

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

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



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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
C1(c(C(x0))) → A1(b(A(x0)))
B1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(x)) → B1(x)
B1(b(A(x0))) → C1(b(A(x0)))
B1(b(A(x0))) → B1(c(b(A(x0))))
B1(a(a(x0))) → A1(a(b(x0)))
C1(c(x)) → B1(c(x))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → A1(b(x))
A1(a(x)) → B1(x)
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
B1(a(A(x0))) → A1(A(x0))
B1(b(a(x0))) → C1(b(a(b(x0))))
A1(a(x)) → B1(a(b(x)))

The TRS R consists of the following rules:

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

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

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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(C(x0))) → A1(b(A(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
B1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(x)) → B1(x)
B1(b(A(x0))) → C1(b(A(x0)))
B1(b(A(x0))) → B1(c(b(A(x0))))
B1(a(a(x0))) → A1(a(b(x0)))
C1(c(x)) → B1(c(x))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → A1(b(x))
A1(a(x)) → B1(x)
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
A1(a(x)) → B1(a(b(x)))
B1(b(a(x0))) → C1(b(a(b(x0))))

The TRS R consists of the following rules:

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

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

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



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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
C1(c(C(x0))) → A1(b(A(x0)))
B1(a(x)) → B1(x)
B1(a(b(x0))) → A1(b(c(a(x0))))
B1(b(A(x0))) → C1(b(A(x0)))
A1(a(b(x0))) → A1(b(c(a(x0))))
B1(b(A(x0))) → B1(c(b(A(x0))))
B1(a(a(x0))) → A1(a(b(x0)))
A1(a(A(x0))) → A1(A(x0))
C1(c(x)) → B1(c(x))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → B1(x)
A1(a(a(x0))) → A1(a(b(x0)))
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
B1(b(a(x0))) → C1(b(a(b(x0))))
A1(a(x)) → B1(a(b(x)))

The TRS R consists of the following rules:

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

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

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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(C(x0))) → A1(b(A(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
B1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(x)) → B1(x)
B1(b(A(x0))) → C1(b(A(x0)))
A1(a(b(x0))) → A1(b(c(a(x0))))
B1(b(A(x0))) → B1(c(b(A(x0))))
B1(a(a(x0))) → A1(a(b(x0)))
C1(c(x)) → B1(c(x))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → B1(x)
A1(a(a(x0))) → A1(a(b(x0)))
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
A1(a(x)) → B1(a(b(x)))
B1(b(a(x0))) → C1(b(a(b(x0))))

The TRS R consists of the following rules:

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

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

B1(b(A(y0))) → B1(c(A(y0)))



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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
C1(c(C(x0))) → A1(b(A(x0)))
B1(a(x)) → B1(x)
B1(a(b(x0))) → A1(b(c(a(x0))))
B1(b(A(x0))) → C1(b(A(x0)))
A1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(a(x0))) → A1(a(b(x0)))
C1(c(x)) → B1(c(x))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → B1(x)
A1(a(a(x0))) → A1(a(b(x0)))
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
B1(b(A(y0))) → B1(c(A(y0)))
B1(b(a(x0))) → C1(b(a(b(x0))))
A1(a(x)) → B1(a(b(x)))

The TRS R consists of the following rules:

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

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

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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(C(x0))) → A1(b(A(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
B1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(x)) → B1(x)
B1(b(A(x0))) → C1(b(A(x0)))
A1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(a(x0))) → A1(a(b(x0)))
C1(c(x)) → B1(c(x))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → B1(x)
A1(a(a(x0))) → A1(a(b(x0)))
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
A1(a(x)) → B1(a(b(x)))
B1(b(a(x0))) → C1(b(a(b(x0))))

The TRS R consists of the following rules:

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

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

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



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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(C(x0))) → B1(A(x0))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
C1(c(C(x0))) → A1(b(A(x0)))
B1(a(x)) → B1(x)
B1(a(b(x0))) → A1(b(c(a(x0))))
C1(c(b(B(x0)))) → B1(c(C(x0)))
B1(b(A(x0))) → C1(b(A(x0)))
A1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(a(x0))) → A1(a(b(x0)))
C1(c(c(x0))) → B1(a(b(c(x0))))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → B1(x)
C1(c(b(x0))) → B1(c(x0))
A1(a(a(x0))) → A1(a(b(x0)))
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
B1(b(a(x0))) → C1(b(a(b(x0))))
A1(a(x)) → B1(a(b(x)))

The TRS R consists of the following rules:

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

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

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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
C1(c(C(x0))) → A1(b(A(x0)))
B1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(x)) → B1(x)
C1(c(b(B(x0)))) → B1(c(C(x0)))
B1(b(A(x0))) → C1(b(A(x0)))
A1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(a(x0))) → A1(a(b(x0)))
C1(c(c(x0))) → B1(a(b(c(x0))))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → B1(x)
C1(c(b(x0))) → B1(c(x0))
A1(a(a(x0))) → A1(a(b(x0)))
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
A1(a(x)) → B1(a(b(x)))
B1(b(a(x0))) → C1(b(a(b(x0))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(C(x0))) → A1(b(A(x0))) at position [0] we obtained the following new rules:

C1(c(C(x0))) → A1(A(x0))



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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
B1(a(x)) → B1(x)
B1(a(b(x0))) → A1(b(c(a(x0))))
C1(c(b(B(x0)))) → B1(c(C(x0)))
B1(b(A(x0))) → C1(b(A(x0)))
A1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(a(x0))) → A1(a(b(x0)))
C1(c(c(x0))) → B1(a(b(c(x0))))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → B1(x)
A1(a(a(x0))) → A1(a(b(x0)))
C1(c(b(x0))) → B1(c(x0))
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
C1(c(C(x0))) → A1(A(x0))
B1(b(a(x0))) → C1(b(a(b(x0))))
A1(a(x)) → B1(a(b(x)))

The TRS R consists of the following rules:

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

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

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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
B1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(x)) → B1(x)
C1(c(b(B(x0)))) → B1(c(C(x0)))
B1(b(A(x0))) → C1(b(A(x0)))
A1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(a(x0))) → A1(a(b(x0)))
C1(c(c(x0))) → B1(a(b(c(x0))))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → B1(x)
A1(a(a(x0))) → A1(a(b(x0)))
C1(c(b(x0))) → B1(c(x0))
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
B1(b(a(x0))) → C1(b(a(b(x0))))
A1(a(x)) → B1(a(b(x)))

The TRS R consists of the following rules:

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

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

C1(c(b(B(x0)))) → B1(A(x0))



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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
C1(c(b(B(x0)))) → B1(A(x0))
B1(a(x)) → B1(x)
B1(a(b(x0))) → A1(b(c(a(x0))))
B1(b(A(x0))) → C1(b(A(x0)))
A1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(a(x0))) → A1(a(b(x0)))
C1(c(c(x0))) → B1(a(b(c(x0))))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → B1(x)
C1(c(b(x0))) → B1(c(x0))
A1(a(a(x0))) → A1(a(b(x0)))
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
A1(a(x)) → B1(a(b(x)))
B1(b(a(x0))) → C1(b(a(b(x0))))

The TRS R consists of the following rules:

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

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

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

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

C1(c(b(x0))) → A1(b(c(x0)))
C1(c(b(B(x0)))) → A1(b(c(C(x0))))
B1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(x)) → B1(x)
B1(b(A(x0))) → C1(b(A(x0)))
A1(a(b(x0))) → A1(b(c(a(x0))))
B1(a(a(x0))) → A1(a(b(x0)))
C1(c(c(x0))) → B1(a(b(c(x0))))
B1(b(a(x0))) → B1(c(b(a(b(x0)))))
A1(a(x)) → B1(x)
C1(c(b(x0))) → B1(c(x0))
A1(a(a(x0))) → A1(a(b(x0)))
C1(b(x)) → C1(x)
C1(c(c(x0))) → A1(b(a(b(c(x0)))))
B1(b(x)) → A1(x)
A1(a(x)) → B1(a(b(x)))
B1(b(a(x0))) → C1(b(a(b(x0))))

The TRS R consists of the following rules:

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

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

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

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

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

The set Q is empty.

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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

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

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

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

Q is empty.