Consider the TRS R consisting of the rewrite rules

1: f(f(X)) -> f(g(f(g(f(X)))))
2: f(g(f(X))) -> f(g(X))

There are 3 dependency pairs:

3: F(f(X)) -> F(g(f(g(f(X)))))
4: F(f(X)) -> F(g(f(X)))
5: F(g(f(X))) -> F(g(X))

The approximated dependency graph contains one SCC:
{5}.

- Consider the SCC {5}.
There are no usable rules.
By taking the polynomial interpretation
[f](x) = [F](x) = [g](x) = x + 1,
rule 5
is strictly decreasing.

Hence the TRS is terminating.