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:

f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)

Q is empty.


↳ QTRS
  ↳ Overlay + Local Confluence

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

f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)

Q is empty.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof

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

f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)

The set Q consists of the following terms:

f(x0, f(a, f(f(a, a), a)))


Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

F(x, f(a, f(f(a, a), a))) → F(f(a, x), x)
F(x, f(a, f(f(a, a), a))) → F(a, x)

The TRS R consists of the following rules:

f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)

The set Q consists of the following terms:

f(x0, f(a, f(f(a, a), a)))

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ QDPOrderProof

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

F(x, f(a, f(f(a, a), a))) → F(f(a, x), x)
F(x, f(a, f(f(a, a), a))) → F(a, x)

The TRS R consists of the following rules:

f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)

The set Q consists of the following terms:

f(x0, f(a, f(f(a, a), a)))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


F(x, f(a, f(f(a, a), a))) → F(a, x)
The remaining pairs can at least be oriented weakly.

F(x, f(a, f(f(a, a), a))) → F(f(a, x), x)
Used ordering: Combined order from the following AFS and order.
F(x1, x2)  =  F(x1, x2)
f(x1, x2)  =  f
a  =  a

Recursive Path Order [2].
Precedence:
[F2, f] > a


The following usable rules [14] were oriented:

f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP

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

F(x, f(a, f(f(a, a), a))) → F(f(a, x), x)

The TRS R consists of the following rules:

f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)

The set Q consists of the following terms:

f(x0, f(a, f(f(a, a), a)))

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