Term Rewriting System R:

   [X, M, N]
   filter(cons(X), 0, M) -> cons(0)
   filter(cons(X), s(N), M) -> cons(X)
   sieve(cons(0)) -> cons(0)
   sieve(cons(s(N))) -> cons(s(N))
   nats(N) -> cons(N)
   zprimes -> sieve(nats(s(s(0))))

Termination of R to be shown.


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

   filter(cons(X), 0, M) -> cons(0)
   filter(cons(X), s(N), M) -> cons(X)

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(sieve(x_1)) = x_1
POL(nats(x_1)) = x_1
POL(zprimes) = 0
POL(filter(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
POL(0) = 0
POL(cons(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: 

   sieve(cons(0)) -> cons(0)
   sieve(cons(s(N))) -> cons(s(N))

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(sieve(x_1)) = 2*x_1
POL(nats(x_1)) = 1 + x_1
POL(zprimes) = 2
POL(0) = 0
POL(cons(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: 

   nats(N) -> cons(N)

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(sieve(x_1)) = x_1
POL(nats(x_1)) = 1 + x_1
POL(zprimes) = 1
POL(0) = 0
POL(cons(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: 

   zprimes -> sieve(nats(s(s(0))))

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(sieve(x_1)) = x_1
POL(nats(x_1)) = x_1
POL(zprimes) = 1
POL(0) = 0
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.439 seconds.