Consider the TRS R consisting of the rewrite rules

1: from(X) -> cons(X,n__from(s(X)))
2: head(cons(X,XS)) -> X
3: 2nd(cons(X,XS)) -> head(activate(XS))
4: take(0,XS) -> nil
5: take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
6: sel(0,cons(X,XS)) -> X
7: sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
8: from(X) -> n__from(X)
9: take(X1,X2) -> n__take(X1,X2)
10: activate(n__from(X)) -> from(X)
11: activate(n__take(X1,X2)) -> take(X1,X2)
12: activate(X) -> X

There are 7 dependency pairs:

13: 2nd#(cons(X,XS)) -> HEAD(activate(XS))
14: 2nd#(cons(X,XS)) -> ACTIVATE(XS)
15: TAKE(s(N),cons(X,XS)) -> ACTIVATE(XS)
16: SEL(s(N),cons(X,XS)) -> SEL(N,activate(XS))
17: SEL(s(N),cons(X,XS)) -> ACTIVATE(XS)
18: ACTIVATE(n__from(X)) -> FROM(X)
19: ACTIVATE(n__take(X1,X2)) -> TAKE(X1,X2)

The approximated dependency graph contains 2 SCCs:
{15,19}
and {16}.

- Consider the SCC {15,19}.
There are no usable rules.
By taking the polynomial interpretation
[ACTIVATE](x) = [s](x) = x + 1
and [cons](x,y) = [n__take](x,y) = [TAKE](x,y) = x + y + 1,
the rules in {15,19}
are strictly decreasing.

- Consider the SCC {16}.
The usable rules are {1,4,5,8-12}.
By taking the polynomial interpretation
[0] = [nil] = 1,
[n__from](x) = x,
[activate](x) = [from](x) = [s](x) = [SEL](x,y) = x + 1,
[n__take](x,y) = [take](x,y) = x + y + 1
and [cons](x,y) = y,
the rules in {1,5,9,10}
are weakly decreasing and
the rules in {4,8,11,12,16}
are strictly decreasing.

Hence the TRS is terminating.