Term Rewriting System R:

   [X, Y, Z]
   f(X) -> cons(X, n__f(g(X)))
   f(X) -> n__f(X)
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   activate(n__f(X)) -> f(X)
   activate(X) -> X

Termination of R to be shown.


R contains the following Dependency Pairs: 


   F(X) -> G(X)
   SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
   SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
   ACTIVATE(n__f(X)) -> F(X)
   G(s(X)) -> G(X)

Furthermore, R contains two SCCs.


SCC1:

G(s(X)) -> G(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(G(x_1)) = 1 + x_1
POL(s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   G(s(X)) -> G(X)



 This transformation is resulting in no new subcycles.


SCC2:

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

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(g(x_1)) = 0
POL(activate(x_1)) = 0
POL(s(x_1)) = 1 + x_1
POL(SEL(x_1, x_2)) = x_1
POL(f(x_1)) = 0
POL(n__f(x_1)) = 0
POL(0) = 0
POL(cons(x_1, x_2)) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.607 seconds.