Term Rewriting System R:

   [X]
   f(0) -> cons(0, n__f(n__s(n__0)))
   f(s(0)) -> f(p(s(0)))
   f(X) -> n__f(X)
   p(s(0)) -> 0
   s(X) -> n__s(X)
   0 -> n__0
   activate(n__f(X)) -> f(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__0) -> 0
   activate(X) -> X

Termination of R to be shown.


Removing the following rules from R which fullfill a polynomial ordering: 

   f(0) -> cons(0, n__f(n__s(n__0)))
   f(X) -> n__f(X)

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(activate(x_1)) = 2*x_1
POL(n__0) = 0
POL(f(x_1)) = 2 + x_1
POL(0) = 0
POL(n__f(x_1)) = 1 + x_1
POL(p(x_1)) = x_1
POL(n__s(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1 + x_2
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   s(X) -> n__s(X)
   p(s(0)) -> 0

where the Polynomial interpretation:
POL(s(x_1)) = 2 + x_1
POL(activate(x_1)) = 2*x_1
POL(n__0) = 0
POL(f(x_1)) = x_1
POL(0) = 0
POL(n__f(x_1)) = x_1
POL(p(x_1)) = x_1
POL(n__s(x_1)) = 1 + x_1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   0 -> n__0
   activate(X) -> X

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(activate(x_1)) = 1 + x_1
POL(n__0) = 0
POL(f(x_1)) = x_1
POL(0) = 1
POL(n__f(x_1)) = x_1
POL(p(x_1)) = x_1
POL(n__s(x_1)) = x_1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   activate(n__f(X)) -> f(activate(X))

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(activate(x_1)) = x_1
POL(n__0) = 0
POL(f(x_1)) = x_1
POL(0) = 0
POL(n__f(x_1)) = 1 + x_1
POL(p(x_1)) = x_1
POL(n__s(x_1)) = x_1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   activate(n__s(X)) -> s(activate(X))

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(activate(x_1)) = x_1
POL(n__0) = 0
POL(f(x_1)) = x_1
POL(0) = 0
POL(p(x_1)) = x_1
POL(n__s(x_1)) = 1 + x_1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   activate(n__0) -> 0

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(activate(x_1)) = 1 + x_1
POL(n__0) = 0
POL(f(x_1)) = x_1
POL(0) = 0
POL(p(x_1)) = x_1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   F(s(0)) -> F(p(s(0)))

R contains no SCCs.




Termination of R successfully shown.


Duration: 0.625 seconds.