Consider the TRS R consisting of the rewrite rules

1: f(0) -> cons(0,n__f(s(0)))
2: f(s(0)) -> f(p(s(0)))
3: p(s(0)) -> 0
4: f(X) -> n__f(X)
5: activate(n__f(X)) -> f(X)
6: activate(X) -> X

There are 3 dependency pairs:

7: F(s(0)) -> F(p(s(0)))
8: F(s(0)) -> P(s(0))
9: ACTIVATE(n__f(X)) -> F(X)

The approximated dependency graph contains one SCC:
{7}.

- Consider the SCC {7}.
The usable rules are {3}.
By taking the polynomial interpretation
[0] = [p](x) = 1
and [F](x) = [s](x) = x + 1,
rule 3
is weakly decreasing and
rule 7
is strictly decreasing.

Hence the TRS is terminating.