Term Rewriting System R:

   [YS, X, XS, Y, L]
   app(nil, YS) -> YS
   app(cons(X, XS), YS) -> cons(X, app(XS, YS))
   from(X) -> cons(X, from(s(X)))
   zWadr(nil, YS) -> nil
   zWadr(XS, nil) -> nil
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS))
   prefix(L) -> cons(nil, zWadr(L, prefix(L)))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   FROM(X) -> FROM(s(X))
   ZWADR(cons(X, XS), cons(Y, YS)) -> APP(Y, cons(X, nil))
   ZWADR(cons(X, XS), cons(Y, YS)) -> ZWADR(XS, YS)
   APP(cons(X, XS), YS) -> APP(XS, YS)
   PREFIX(L) -> ZWADR(L, prefix(L))
   PREFIX(L) -> PREFIX(L)

Furthermore, R contains four SCCs.


SCC1:

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

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)) = x_1
POL(FROM(x_1)) = 1 + x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   from(X) -> cons(X, from(s(X)))
   zWadr(XS, nil) -> nil
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS))
   zWadr(nil, YS) -> nil
   app(nil, YS) -> YS
   app(cons(X, XS), YS) -> cons(X, app(XS, YS))
   prefix(L) -> cons(nil, zWadr(L, prefix(L)))


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

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

Applying the non-overlappingness check to the DPs.

The transformation is resulting in one subcycle:

SCC1.MRR1.NOC1:

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: 0.593 seconds.