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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pair:

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


Rules:


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





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
AFTER(x1, x2) -> x1
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes