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: bits(0) -> 0
5: bits(s(x)) -> s(bits(half(s(x))))

There are 3 dependency pairs:

6: HALF(s(s(x))) -> HALF(x)
7: BITS(s(x)) -> BITS(half(s(x)))
8: BITS(s(x)) -> HALF(s(x))

The approximated dependency graph contains 2 SCCs:
{6}
and {7}.

- Consider the SCC {6}.
There are no usable rules.
By taking the polynomial interpretation
[HALF](x) = [s](x) = x + 1,
rule 6
is strictly decreasing.

- Consider the SCC {7}.
The usable rules are {1-3}.
By taking the polynomial interpretation
[0] = 0,
[BITS](x) = x,
[s](x) = x + 1
and [half](x) = x - 1,
we obtain a quasi-model of the usable rules.
Furthermore, dependency pair 7
is strictly decreasing.

Hence the TRS is terminating.