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

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

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

Q is empty.

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

The TRS R consists of the following rules:

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

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

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ EdgeDeletionProof

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
QDP
          ↳ DependencyGraphProof

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 5 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
QDP

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

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

The TRS R consists of the following rules:

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

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