Term Rewriting System R:
[X, XS, N]
from(X) -> cons(X, from(s(X)))
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, XS)

Termination of R to be shown.



   R
Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.


   R
OC
       →TRS2
Dependency Pair Analysis



R contains the following Dependency Pairs:

FROM(X) -> FROM(s(X))
AFTER(s(N), cons(X, XS)) -> AFTER(N, XS)

Furthermore, R contains two SCCs.


   R
OC
       →TRS2
DPs
           →DP Problem 1
Usable Rules (Innermost)


Dependency Pair:

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


Rules:


from(X) -> cons(X, from(s(X)))
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, XS)


Strategy:

innermost




As we are in the innermost case, we can delete all 3 non-usable-rules.


   R
OC
       →TRS2
DPs
           →DP Problem 1
UsableRules
             ...
               →DP Problem 3
Non Termination


Dependency Pair:

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


Rule:

none


Strategy:

innermost




Found an infinite P-chain over R:
P =

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

R = none

s = FROM(X)
evaluates to t =FROM(s(X))

Thus, s starts an infinite chain as s matches t.

Non-Termination of R could be shown.
Duration:
0:02 minutes