Term Rewriting System R:

   [x, y]
   average(s(x), y) -> average(x, s(y))
   average(x, s(s(s(y)))) -> s(average(s(x), y))
   average(0, 0) -> 0
   average(0, s(0)) -> 0
   average(0, s(s(0))) -> s(0)

Termination of R to be shown.


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

   average(s(x), y) -> average(x, s(y))
   average(0, s(0)) -> 0
   average(0, s(s(0))) -> s(0)

where the Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(0) = 0
POL(average(x_1, x_2)) = 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: 

   average(x, s(s(s(y)))) -> s(average(s(x), y))

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

   average(0, 0) -> 0

where the Polynomial interpretation:
POL(0) = 0
POL(average(x_1, x_2)) = 1 + x_1 + x_2
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.547 seconds.