Term Rewriting System R:

   [X, Y, Z]
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   s(X) -> n__s(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

Termination of R to be shown.


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(n__s(X)) -> S(activate(X))
   ACTIVATE(n__s(X)) -> ACTIVATE(X)
   ACTIVATE(n__from(X)) -> FROM(activate(X))
   ACTIVATE(n__from(X)) -> ACTIVATE(X)

Furthermore, R contains two SCCs.


SCC1:

ACTIVATE(n__from(X)) -> ACTIVATE(X)
ACTIVATE(n__s(X)) -> ACTIVATE(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(n__from(x_1)) = 1 + x_1
POL(ACTIVATE(x_1)) = 1 + x_1
POL(n__s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   ACTIVATE(n__from(X)) -> ACTIVATE(X)
   ACTIVATE(n__s(X)) -> ACTIVATE(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(n__from(x_1)) = 0
POL(activate(x_1)) = 0
POL(s(x_1)) = 1 + x_1
POL(SEL(x_1, x_2)) = x_1
POL(from(x_1)) = 0
POL(n__s(x_1)) = 0
POL(cons(x_1, x_2)) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.619 seconds.