Term Rewriting System R:

   [X, Y]
   f(X) -> if(X, c, n__f(true))
   f(X) -> n__f(X)
   if(true, X, Y) -> X
   if(false, X, Y) -> activate(Y)
   activate(n__f(X)) -> f(X)
   activate(X) -> X

Termination of R to be shown.


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

   f(X) -> if(X, c, n__f(true))
   f(X) -> n__f(X)
   activate(X) -> X

where the Polynomial interpretation:
POL(activate(x_1)) = 1 + x_1
POL(true) = 0
POL(f(x_1)) = 1 + x_1
POL(c) = 0
POL(n__f(x_1)) = x_1
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(false) = 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: 

   if(true, X, Y) -> X

where the Polynomial interpretation:
POL(activate(x_1)) = x_1
POL(true) = 1
POL(f(x_1)) = x_1
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(n__f(x_1)) = x_1
POL(false) = 0
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: 

   if(false, X, Y) -> activate(Y)

where the Polynomial interpretation:
POL(activate(x_1)) = x_1
POL(f(x_1)) = x_1
POL(n__f(x_1)) = x_1
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(false) = 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(X)

where the Polynomial interpretation:
POL(activate(x_1)) = 1 + x_1
POL(f(x_1)) = x_1
POL(n__f(x_1)) = x_1
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.405 seconds.