Consider the TRS R consisting of the rewrite rules

1: sum(0) -> 0
2: sum(s(x)) -> sum(x) + s(x)
3: x + 0 -> x
4: x + s(y) -> s(x + y)

There are 3 dependency pairs:

5: SUM(s(x)) -> sum(x) +# s(x)
6: SUM(s(x)) -> SUM(x)
7: x +# s(y) -> x +# y

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

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

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

Hence the TRS is terminating.