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

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

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

Q is empty.

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.

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

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

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

The TRS R consists of the following rules:

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

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

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



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

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

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

The TRS R consists of the following rules:

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

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

A(b(c(x0))) → A(x0)



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

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

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

The TRS R consists of the following rules:

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

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