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

Q is empty.

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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


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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

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

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

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

C(a(b(x))) → C(c(x))
C(a(B(x))) → B1(B(x))
A1(x) → B1(x)
C(a(b(x))) → A1(b(c(c(x))))
C(a(B(x))) → A1(A(x))
C(a(B(x))) → A2(x)
C(a(B(x))) → A1(b(B(x)))
C(a(b(x))) → A1(a(b(c(c(x)))))
C(a(B(x))) → B1(a(B(x)))
C(a(b(x))) → C(x)
C(a(b(x))) → B1(c(c(x)))

The TRS R consists of the following rules:

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

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ 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:

C(a(b(x))) → C(c(x))
C(a(B(x))) → B1(B(x))
A1(x) → B1(x)
C(a(b(x))) → A1(b(c(c(x))))
C(a(B(x))) → A1(A(x))
C(a(B(x))) → A2(x)
C(a(B(x))) → A1(b(B(x)))
C(a(b(x))) → A1(a(b(c(c(x)))))
C(a(B(x))) → B1(a(B(x)))
C(a(b(x))) → C(x)
C(a(b(x))) → B1(c(c(x)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 9 less nodes.

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

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

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

The TRS R consists of the following rules:

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

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

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

The set Q is empty.

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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

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

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

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

Q is empty.

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

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

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

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

The set Q is empty.

↳ QTRS
  ↳ DependencyPairsProof
  ↳ QTRS Reverse
QTRS
  ↳ QTRS Reverse

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

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

Q is empty.

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

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

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

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

The set Q is empty.

↳ QTRS
  ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse
QTRS

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

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

Q is empty.