Term Rewriting System R:

   [x]
   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(half(x))))

Termination of R to be shown.


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

   log(s(0)) -> 0

where the Polynomial interpretation:
POL(log(x_1)) = 1 + x_1
POL(s(x_1)) = x_1
POL(half(x_1)) = x_1
POL(0) = 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: 

   half(s(s(x))) -> s(half(x))

where the Polynomial interpretation:
POL(log(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(half(x_1)) = x_1
POL(0) = 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: 

   log(s(s(x))) -> s(log(s(half(x))))

where the Polynomial interpretation:
POL(log(x_1)) = 2*x_1
POL(s(x_1)) = 1 + x_1
POL(half(x_1)) = x_1
POL(0) = 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: 

   half(0) -> 0

where the Polynomial interpretation:
POL(half(x_1)) = 1 + x_1
POL(0) = 0
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 1.79 seconds.