Termination w.r.t. Q proof of /tmp/tmpb1N2tC/jw14.trs

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

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

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f(a, f(a, x)) → f(a, f(f(a, a), f(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:

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

The TRS R consists of the following rules:

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

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:

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

The TRS R consists of the following rules:

f(a, f(a, x)) → f(a, f(f(a, a), f(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 0 SCCs with 3 less nodes.