Term Rewriting System R:

   [X, Y, Z]
   and(true, X) -> X
   and(false, Y) -> false
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, first(X, 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))
   ADD(s(X), Y) -> ADD(X, Y)
   FIRST(s(X), cons(Y, Z)) -> FIRST(X, Z)

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: 0.573 seconds.