Consider the TRS R consisting of the rewrite rules

1: half(0) -> 0
2: half(s(0)) -> 0
3: half(s(s(x))) -> s(half(x))
4: s(log(0)) -> s(0)
5: log(s(x)) -> s(log(half(s(x))))

There are 6 dependency pairs:

6: HALF(s(s(x))) -> S(half(x))
7: HALF(s(s(x))) -> HALF(x)
8: S(log(0)) -> S(0)
9: LOG(s(x)) -> S(log(half(s(x))))
10: LOG(s(x)) -> LOG(half(s(x)))
11: LOG(s(x)) -> HALF(s(x))

The approximated dependency graph contains 2 SCCs:
{7}
and {10}.

- Consider the SCC {7}.
There are no usable rules.
By taking the polynomial interpretation
[HALF](x) = [s](x) = x + 1,
rule 7
is strictly decreasing.

- Consider the SCC {10}.
The usable rules are {1-4}.
By taking the polynomial interpretation
[0] = [log](x) = 0,
[LOG](x) = x,
[s](x) = x + 1
and [half](x) = x - 1,
we obtain a quasi-model of the usable rules.
Furthermore, dependency pair 10
is strictly decreasing.

Hence the TRS is terminating.