Term Rewriting System R:
[X, Y, Z]
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nfrom(X)) -> from(X)
activate(X) -> X

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nfrom(X)) -> FROM(X)

Furthermore, R contains one SCC. 


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


Dependency Pair:

SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))


Rules:


from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nfrom(X)) -> from(X)
activate(X) -> X





The following dependency pair can be strictly oriented:

SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))


The following usable rules using the Ce-refinement can be oriented:

activate(nfrom(X)) -> from(X)
activate(X) -> X
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
SEL > activate > from > nfrom
SEL > activate > from > cons
SEL > activate > from > s

resulting in one new DP problem.
Used Argument Filtering System:
SEL(x1, x2) -> SEL(x1, x2)
s(x1) -> s(x1)
cons(x1, x2) -> cons(x1, x2)
activate(x1) -> activate(x1)
nfrom(x1) -> nfrom(x1)
from(x1) -> from(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)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
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