Consider the TRS R consisting of the rewrite rules

1: f(h(x)) -> f(i(x))
2: g(i(x)) -> g(h(x))
3: h(a) -> b
4: i(a) -> b

There are 4 dependency pairs:

5: F(h(x)) -> F(i(x))
6: F(h(x)) -> I(x)
7: G(i(x)) -> G(h(x))
8: G(i(x)) -> H(x)

The approximated dependency graph contains 2 SCCs:
{5}
and {7}.

- Consider the SCC {5}.
The usable rules are {4}.
By taking the polynomial interpretation
[a] = [b] = 1,
[i](x) = x
and [F](x) = [h](x) = x + 1,
rule 4
is weakly decreasing and
rule 5
is strictly decreasing.

- Consider the SCC {7}.
The usable rules are {3}.
By taking the polynomial interpretation
[a] = [b] = 1,
[h](x) = x
and [G](x) = [i](x) = x + 1,
rule 3
is weakly decreasing and
rule 7
is strictly decreasing.

Hence the TRS is terminating.