Term Rewriting System R:

   [x, y]
   f(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

Termination of R to be shown.


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

   f(0) -> s(0)
   f(s(0)) -> s(s(0))

where the Polynomial interpretation:
POL(*(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = x_1
POL(f(x_1)) = 1 + x_1
POL(+(x_1, x_2)) = 1 + x_1 + x_2
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: 

   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

where the Polynomial interpretation:
POL(*(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = x_1
POL(f(x_1)) = 2*x_1
POL(+(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
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(0)

R contains no SCCs.




Termination of R successfully shown.


Duration: 0.507 seconds.