Term Rewriting System R:

   [X, Z, N, Y]
   from(X) -> cons(X, from(s(X)))
   2ndspos(0, Z) -> rnil
   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
   2ndsneg(0, Z) -> rnil
   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
   pi(X) -> 2ndspos(X, from(0))
   plus(0, Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   times(0, Y) -> 0
   times(s(X), Y) -> plus(Y, times(X, Y))
   square(X) -> times(X, 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: 


   TIMES(s(X), Y) -> PLUS(Y, times(X, Y))
   TIMES(s(X), Y) -> TIMES(X, Y)
   PLUS(s(X), Y) -> PLUS(X, Y)
   FROM(X) -> FROM(s(X))
   2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, Z))
   2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, Z)
   SQUARE(X) -> TIMES(X, X)
   2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, Z))
   2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, Z)
   PI(X) -> 2NDSPOS(X, from(0))
   PI(X) -> FROM(0)

Furthermore, R contains four SCCs.


SCC1:

PLUS(s(X), Y) -> PLUS(X, Y)

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(PLUS(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   PLUS(s(X), Y) -> PLUS(X, Y)



 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: 0.872 seconds.