Term Rewriting System R:

   [X, Y, Z]
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   dbls(nil) -> nil
   dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)
   indx(nil, X) -> nil
   indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z))
   from(X) -> cons(X, from(s(X)))

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))
   INDX(cons(X, Y), Z) -> SEL(X, Z)
   INDX(cons(X, Y), Z) -> INDX(Y, Z)
   DBLS(cons(X, Y)) -> DBL(X)
   DBLS(cons(X, Y)) -> DBLS(Y)
   DBL(s(X)) -> DBL(X)
   SEL(s(X), cons(Y, Z)) -> SEL(X, Z)

Furthermore, R contains five 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: 0.623 seconds.