Term Rewriting System R:

   [N, X, Y]
   terms(N) -> cons(recip(sqr(N)))
   sqr(0) -> 0
   sqr(s) -> s
   dbl(0) -> 0
   dbl(s) -> s
   add(0, X) -> X
   add(s, Y) -> s
   first(0, X) -> nil
   first(s, cons(Y)) -> cons(Y)

Termination of R to be shown.


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

   terms(N) -> cons(recip(sqr(N)))

where the Polynomial interpretation:
POL(add(x_1, x_2)) = x_1 + x_2
POL(first(x_1, x_2)) = x_1 + x_2
POL(nil) = 0
POL(s) = 0
POL(dbl(x_1)) = x_1
POL(terms(x_1)) = 1 + x_1
POL(sqr(x_1)) = x_1
POL(recip(x_1)) = x_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: 

   sqr(0) -> 0
   sqr(s) -> s

where the Polynomial interpretation:
POL(add(x_1, x_2)) = x_1 + x_2
POL(first(x_1, x_2)) = x_1 + x_2
POL(nil) = 0
POL(s) = 0
POL(dbl(x_1)) = x_1
POL(sqr(x_1)) = 1 + x_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: 

   dbl(0) -> 0
   dbl(s) -> s

where the Polynomial interpretation:
POL(add(x_1, x_2)) = x_1 + x_2
POL(first(x_1, x_2)) = x_1 + x_2
POL(nil) = 0
POL(dbl(x_1)) = 1 + x_1
POL(s) = 0
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: 

   add(0, X) -> X
   add(s, Y) -> s

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

   first(0, X) -> nil

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

   first(s, cons(Y)) -> cons(Y)

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



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.436 seconds.