Term Rewriting System R:

   [X]
   f(X) -> g(n__h(n__f(X)))
   f(X) -> n__f(X)
   h(X) -> n__h(X)
   activate(n__h(X)) -> h(activate(X))
   activate(n__f(X)) -> f(activate(X))
   activate(X) -> X

Termination of R to be shown.


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

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

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(activate(x_1)) = 2*x_1
POL(h(x_1)) = x_1
POL(f(x_1)) = 1 + x_1
POL(n__f(x_1)) = 1 + x_1
POL(n__h(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: 

   f(X) -> g(n__h(n__f(X)))
   f(X) -> n__f(X)

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(activate(x_1)) = x_1
POL(h(x_1)) = x_1
POL(f(x_1)) = 1 + x_1
POL(n__f(x_1)) = x_1
POL(n__h(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__h(X)) -> h(activate(X))

where the Polynomial interpretation:
POL(activate(x_1)) = 2*x_1
POL(h(x_1)) = 1 + x_1
POL(n__h(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(X) -> X

where the Polynomial interpretation:
POL(activate(x_1)) = 1 + x_1
POL(h(x_1)) = x_1
POL(n__h(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: 

   h(X) -> n__h(X)

where the Polynomial interpretation:
POL(h(x_1)) = 1 + x_1
POL(n__h(x_1)) = x_1
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.459 seconds.