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

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ DependencyGraphProof

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

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

The TRS R consists of the following rules:

a(x1) → x1
a(b(x1)) → b(b(c(a(x1))))
b(x1) → x1
c(c(b(x1))) → 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 2 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
QDP
          ↳ Narrowing

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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