Consider the TRS R consisting of the rewrite rules

1: f(0) -> cons(0,n__f(n__s(n__0)))
2: f(s(0)) -> f(p(s(0)))
3: p(s(0)) -> 0
4: f(X) -> n__f(X)
5: s(X) -> n__s(X)
6: 0 -> n__0
7: activate(n__f(X)) -> f(activate(X))
8: activate(n__s(X)) -> s(activate(X))
9: activate(n__0) -> 0
10: activate(X) -> X

There are 7 dependency pairs:

11: F(s(0)) -> F(p(s(0)))
12: F(s(0)) -> P(s(0))
13: ACTIVATE(n__f(X)) -> F(activate(X))
14: ACTIVATE(n__f(X)) -> ACTIVATE(X)
15: ACTIVATE(n__s(X)) -> S(activate(X))
16: ACTIVATE(n__s(X)) -> ACTIVATE(X)
17: ACTIVATE(n__0) -> 0#

The approximated dependency graph contains 2 SCCs:
{11}
and {14,16}.

- Consider the SCC {11}.
The usable rules are {3,5,6}.
By taking the polynomial interpretation
[0] = [n__0] = [p](x) = 1
and [F](x) = [n__s](x) = [s](x) = x + 1,
the rules in {3,5,6}
are weakly decreasing and
rule 11
is strictly decreasing.

- Consider the SCC {14,16}.
There are no usable rules.
By taking the polynomial interpretation
[ACTIVATE](x) = [n__f](x) = [n__s](x) = x + 1,
the rules in {14,16}
are strictly decreasing.

Hence the TRS is terminating.