Term Rewriting System R:

   [X, Z, Y, X1, X2]
   from(X) -> cons(X, n__from(s(X)))
   from(X) -> n__from(X)
   first(0, Z) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   first(X1, X2) -> n__first(X1, X2)
   sel(0, cons(X, Z)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   activate(n__from(X)) -> from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X

Termination of R to be shown.


R contains the following Dependency Pairs: 


   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
   SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
   SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
   ACTIVATE(n__from(X)) -> FROM(X)
   ACTIVATE(n__first(X1, X2)) -> FIRST(X1, X2)

Furthermore, R contains two SCCs.


SCC1:

ACTIVATE(n__first(X1, X2)) -> FIRST(X1, X2)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

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__first(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = x_1
POL(ACTIVATE(x_1)) = x_1
POL(FIRST(x_1, x_2)) = x_1 + x_2
POL(cons(x_1, x_2)) = x_1 + x_2

The following Dependency Pairs can be deleted:

   ACTIVATE(n__first(X1, X2)) -> FIRST(X1, X2)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)



 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(n__first(x_1, x_2)) = 0
POL(first(x_1, x_2)) = 0
POL(nil) = 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(0) = 0
POL(cons(x_1, x_2)) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.693 seconds.