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

Q is empty.


QTRS
  ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

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

Q is empty.

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

A(c(x1)) → A(x1)
A(c(x1)) → A(a(x1))

The TRS R consists of the following rules:

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

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

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

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

A(c(x1)) → A(x1)
A(c(x1)) → A(a(x1))

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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


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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

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

Q is empty.

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

C(c(A(x0))) → B(A(x0))



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

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

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

The TRS R consists of the following rules:

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

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

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

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

The TRS R consists of the following rules:

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

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

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



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

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

The TRS R consists of the following rules:

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The set Q is empty.

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

Q is empty.

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

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

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

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

The set Q is empty.

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

Q is empty.

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

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

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

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

Q is empty.

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

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

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

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

Q is empty.