Term Rewriting System R:

   [X, Y, Z, X1]
   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   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: 


   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)
   2ND(cons(X, X1)) -> 2ND(cons1(X, activate(X1)))
   2ND(cons(X, X1)) -> ACTIVATE(X1)

Furthermore, R contains one SCC.


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.




Termination of R successfully shown.


Duration: 0.550 seconds.