Term Rewriting System R:

   [X, XS, N]
   from(X) -> cons(X, from(s(X)))
   head(cons(X, XS)) -> X
   2nd(cons(X, XS)) -> head(XS)
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   FROM(X) -> FROM(s(X))
   TAKE(s(N), cons(X, XS)) -> TAKE(N, XS)
   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
   2ND(cons(X, XS)) -> HEAD(XS)

Furthermore, R contains three SCCs.


SCC1:

FROM(X) -> FROM(s(X))

Found an infinite P-chain over R:
P = FROM(X) -> FROM(s(X))
R = []
s = FROM(X) evaluates to t = FROM(s(X))
Thus, s starts an infinite reduction as s matches t.


Non-Termination of R could be shown.

Duration: 0.547 seconds.