Term Rewriting System R:

   [X, Y, Z, X1, X2]
   2nd(cons(X, n__cons(Y, Z))) -> activate(Y)
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   cons(X1, X2) -> n__cons(X1, X2)
   s(X) -> n__s(X)
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   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: 


   ACTIVATE(n__s(X)) -> S(activate(X))
   ACTIVATE(n__s(X)) -> ACTIVATE(X)
   ACTIVATE(n__cons(X1, X2)) -> CONS(activate(X1), X2)
   ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__from(X)) -> FROM(activate(X))
   ACTIVATE(n__from(X)) -> ACTIVATE(X)
   2ND(cons(X, n__cons(Y, Z))) -> ACTIVATE(Y)
   FROM(X) -> CONS(X, n__from(n__s(X)))

Furthermore, R contains one SCC.


SCC1:

ACTIVATE(n__from(X)) -> ACTIVATE(X)
ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1)
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__cons(x_1, x_2)) = 1 + x_1 + x_2
POL(n__s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   ACTIVATE(n__from(X)) -> ACTIVATE(X)
   ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__s(X)) -> ACTIVATE(X)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.600 seconds.