Term Rewriting System R:

   [X, Y]
   first(0, X) -> nil
   first(s(X), cons(Y)) -> cons(Y)
   from(X) -> cons(X)

Termination of R to be shown.


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(x_1)) = x_1
POL(from(x_1)) = x_1
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(X), cons(Y)) -> cons(Y)

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

   from(X) -> cons(X)

where the Polynomial interpretation:
POL(from(x_1)) = 1 + x_1
POL(cons(x_1)) = x_1
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.378 seconds.