Consider the TRS R consisting of the rewrite rules

1: g(c(x,s(y))) -> g(c(s(x),y))
2: f(c(s(x),y)) -> f(c(x,s(y)))
3: f(f(x)) -> f(d(f(x)))
4: f(x) -> x

There are 3 dependency pairs:

5: G(c(x,s(y))) -> G(c(s(x),y))
6: F(c(s(x),y)) -> F(c(x,s(y)))
7: F(f(x)) -> F(d(f(x)))

The approximated dependency graph contains 2 SCCs:
{6}
and {5}.

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

- Consider the SCC {5}.
There are no usable rules.
By taking the polynomial interpretation
[G](x) = [s](x) = x + 1
and [c](x,y) = y + 1,
rule 5
is strictly decreasing.

Hence the TRS is terminating.