Consider the TRS R consisting of the rewrite rules

1: from(X) -> cons(X,n__from(s(X)))
2: sel(0,cons(X,XS)) -> X
3: sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
4: minus(X,0) -> 0
5: minus(s(X),s(Y)) -> minus(X,Y)
6: quot(0,s(Y)) -> 0
7: quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
8: zWquot(XS,nil) -> nil
9: zWquot(nil,XS) -> nil
10: zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
11: from(X) -> n__from(X)
12: zWquot(X1,X2) -> n__zWquot(X1,X2)
13: activate(n__from(X)) -> from(X)
14: activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
15: activate(X) -> X

There are 10 dependency pairs:

16: SEL(s(N),cons(X,XS)) -> SEL(N,activate(XS))
17: SEL(s(N),cons(X,XS)) -> ACTIVATE(XS)
18: MINUS(s(X),s(Y)) -> MINUS(X,Y)
19: QUOT(s(X),s(Y)) -> QUOT(minus(X,Y),s(Y))
20: QUOT(s(X),s(Y)) -> MINUS(X,Y)
21: ZWQUOT(cons(X,XS),cons(Y,YS)) -> QUOT(X,Y)
22: ZWQUOT(cons(X,XS),cons(Y,YS)) -> ACTIVATE(XS)
23: ZWQUOT(cons(X,XS),cons(Y,YS)) -> ACTIVATE(YS)
24: ACTIVATE(n__from(X)) -> FROM(X)
25: ACTIVATE(n__zWquot(X1,X2)) -> ZWQUOT(X1,X2)

The approximated dependency graph contains 4 SCCs:
{18},
{19},
{22,23,25}
and {16}.

- Consider the SCC {18}.
There are no usable rules.
By taking the polynomial interpretation
[s](x) = x + 1
and [MINUS](x,y) = x + y + 1,
rule 18
is strictly decreasing.

- Consider the SCC {19}.
The usable rules are {4,5}.
By taking the polynomial interpretation
[0] = 0,
[minus](x,y) = x,
[s](x) = x + 1
and [QUOT](x,y) = x + y + 1,
rule 4
is weakly decreasing and
the rules in {5,19}
are strictly decreasing.

- Consider the SCC {22,23,25}.
There are no usable rules.
By taking the polynomial interpretation
[ACTIVATE](x) = x + 1
and [cons](x,y) = [n__zWquot](x,y) = [ZWQUOT](x,y) = x + y + 1,
the rules in {22,23,25}
are strictly decreasing.

- Consider the SCC {16}.
The usable rules are {1,4-15}.
By taking the polynomial interpretation
[0] = 0,
[nil] = 1,
[minus](x,y) = x,
[activate](x) = [cons](x,y) = [from](x) = [n__from](x) = [s](x) = [SEL](x,y) = x + 1
and [n__zWquot](x,y) = [quot](x,y) = [zWquot](x,y) = x + y + 1,
the rules in {1,4,7,11,12}
are weakly decreasing and
the rules in {5,6,8-10,13-16}
are strictly decreasing.

Hence the TRS is terminating.