Term Rewriting System R:

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

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: 


   FIRST(s(X), cons(Y, Z)) -> FIRST(X, Z)
   FROM(X) -> FROM(s(X))
   SEL(s(X), cons(Y, Z)) -> SEL(X, Z)

Furthermore, R contains three SCCs.


SCC1:

FIRST(s(X), cons(Y, Z)) -> FIRST(X, Z)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(FIRST(x_1, x_2)) = 1 + x_1 + x_2
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   FIRST(s(X), cons(Y, Z)) -> FIRST(X, Z)



 This transformation is resulting in no new subcycles.


SCC2:

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: 0.523 seconds.