Consider the TRS R consisting of the rewrite rules

1: 2nd(cons1(X,cons(Y,Z))) -> Y
2: 2nd(cons(X,X1)) -> 2nd(cons1(X,activate(X1)))
3: from(X) -> cons(X,n__from(n__s(X)))
4: from(X) -> n__from(X)
5: s(X) -> n__s(X)
6: activate(n__from(X)) -> from(activate(X))
7: activate(n__s(X)) -> s(activate(X))
8: activate(X) -> X

There are 6 dependency pairs:

9: 2nd#(cons(X,X1)) -> 2nd#(cons1(X,activate(X1)))
10: 2nd#(cons(X,X1)) -> ACTIVATE(X1)
11: ACTIVATE(n__from(X)) -> FROM(activate(X))
12: ACTIVATE(n__from(X)) -> ACTIVATE(X)
13: ACTIVATE(n__s(X)) -> S(activate(X))
14: ACTIVATE(n__s(X)) -> ACTIVATE(X)

The approximated dependency graph contains one SCC:
{12,14}.

- Consider the SCC {12,14}.
There are no usable rules.
By taking the polynomial interpretation
[ACTIVATE](x) = [n__from](x) = [n__s](x) = x + 1,
the rules in {12,14}
are strictly decreasing.

Hence the TRS is terminating.