Consider the TRS R consisting of the rewrite rules

1: f(0) -> s(0)
2: f(s(0)) -> s(0)
3: f(s(s(x))) -> f(f(s(x)))

There are 2 dependency pairs:

4: F(s(s(x))) -> F(f(s(x)))
5: F(s(s(x))) -> F(s(x))

The approximated dependency graph contains one SCC:
{4,5}.

- Consider the SCC {4,5}.
By taking the polynomial interpretation
[0] = 1
and [f](x) = [F](x) = [s](x) = x + 1,
the rules in {1,3,4}
are weakly decreasing and
the rules in {2,5}
are strictly decreasing.
There is one new SCC.

- Consider the SCC {4}.
By taking the polynomial interpretation
[0] = 0,
[f](x) = 1
and [F](x) = [s](x) = x + 1,
the rules in {1-3}
are weakly decreasing and
rule 4
is strictly decreasing.


Hence the TRS is terminating.