Term Rewriting System R:

   [Z, X, Y, X1, X2]
   fst(0, Z) -> nil
   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
   fst(X1, X2) -> n__fst(X1, X2)
   from(X) -> cons(X, n__from(s(X)))
   from(X) -> n__from(X)
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   add(X1, X2) -> n__add(X1, X2)
   len(nil) -> 0
   len(cons(X, Z)) -> s(n__len(activate(Z)))
   len(X) -> n__len(X)
   activate(n__fst(X1, X2)) -> fst(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__len(X)) -> len(X)
   activate(X) -> X

Termination of R to be shown.


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

   fst(0, Z) -> nil
   add(0, X) -> X
   len(nil) -> 0

where the Polynomial interpretation:
POL(add(x_1, x_2)) = 2*x_1 + x_2
POL(nil) = 1
POL(s(x_1)) = x_1
POL(activate(x_1)) = 2*x_1
POL(fst(x_1, x_2)) = 2*x_1 + 2*x_2
POL(n__fst(x_1, x_2)) = x_1 + x_2
POL(0) = 1
POL(cons(x_1, x_2)) = x_1 + x_2
POL(n__from(x_1)) = x_1
POL(from(x_1)) = 2*x_1
POL(len(x_1)) = 2*x_1
POL(n__len(x_1)) = x_1
POL(n__add(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: 

   fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z)))
   fst(X1, X2) -> n__fst(X1, X2)

where the Polynomial interpretation:
POL(n__from(x_1)) = x_1
POL(add(x_1, x_2)) = 2*x_1 + x_2
POL(activate(x_1)) = 2*x_1
POL(s(x_1)) = x_1
POL(from(x_1)) = 2*x_1
POL(fst(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
POL(n__fst(x_1, x_2)) = 1 + x_1 + x_2
POL(len(x_1)) = 2*x_1
POL(n__len(x_1)) = x_1
POL(n__add(x_1, x_2)) = x_1 + x_2
POL(cons(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: 

   activate(n__fst(X1, X2)) -> fst(X1, X2)

where the Polynomial interpretation:
POL(n__from(x_1)) = x_1
POL(add(x_1, x_2)) = 2*x_1 + x_2
POL(activate(x_1)) = 2*x_1
POL(s(x_1)) = x_1
POL(from(x_1)) = 2*x_1
POL(len(x_1)) = 2*x_1
POL(n__fst(x_1, x_2)) = 1 + x_1 + x_2
POL(fst(x_1, x_2)) = x_1 + x_2
POL(n__len(x_1)) = x_1
POL(n__add(x_1, x_2)) = x_1 + x_2
POL(cons(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: 

   activate(n__from(X)) -> from(X)

where the Polynomial interpretation:
POL(add(x_1, x_2)) = 2*x_1 + x_2
POL(n__from(x_1)) = 1 + x_1
POL(activate(x_1)) = 2*x_1
POL(s(x_1)) = x_1
POL(from(x_1)) = 1 + 2*x_1
POL(len(x_1)) = 2*x_1
POL(n__len(x_1)) = x_1
POL(n__add(x_1, x_2)) = x_1 + x_2
POL(cons(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: 

   activate(n__add(X1, X2)) -> add(X1, X2)

where the Polynomial interpretation:
POL(add(x_1, x_2)) = 1 + 2*x_1 + x_2
POL(n__from(x_1)) = x_1
POL(activate(x_1)) = 2*x_1
POL(s(x_1)) = x_1
POL(from(x_1)) = 2*x_1
POL(len(x_1)) = 2*x_1
POL(n__len(x_1)) = x_1
POL(n__add(x_1, x_2)) = 1 + x_1 + x_2
POL(cons(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: 

   activate(n__len(X)) -> len(X)

where the Polynomial interpretation:
POL(add(x_1, x_2)) = 2*x_1 + x_2
POL(n__from(x_1)) = x_1
POL(activate(x_1)) = 2*x_1
POL(s(x_1)) = x_1
POL(from(x_1)) = 2*x_1
POL(len(x_1)) = 1 + 2*x_1
POL(n__len(x_1)) = 1 + x_1
POL(n__add(x_1, x_2)) = x_1 + x_2
POL(cons(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: 

   activate(X) -> X
   len(X) -> n__len(X)
   add(X1, X2) -> n__add(X1, X2)

where the Polynomial interpretation:
POL(add(x_1, x_2)) = 1 + x_1 + x_2
POL(n__from(x_1)) = x_1
POL(s(x_1)) = x_1
POL(activate(x_1)) = 1 + x_1
POL(from(x_1)) = 2*x_1
POL(len(x_1)) = 1 + x_1
POL(n__len(x_1)) = x_1
POL(n__add(x_1, x_2)) = x_1 + x_2
POL(cons(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: 

   len(cons(X, Z)) -> s(n__len(activate(Z)))

where the Polynomial interpretation:
POL(add(x_1, x_2)) = x_1 + x_2
POL(n__from(x_1)) = x_1
POL(s(x_1)) = x_1
POL(activate(x_1)) = x_1
POL(from(x_1)) = 2*x_1
POL(len(x_1)) = 1 + x_1
POL(n__len(x_1)) = x_1
POL(n__add(x_1, x_2)) = x_1 + x_2
POL(cons(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: 

   add(s(X), Y) -> s(n__add(activate(X), Y))

where the Polynomial interpretation:
POL(n__from(x_1)) = x_1
POL(add(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = x_1
POL(activate(x_1)) = x_1
POL(from(x_1)) = 2*x_1
POL(n__add(x_1, x_2)) = x_1 + x_2
POL(cons(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: 

   from(X) -> cons(X, n__from(s(X)))
   from(X) -> n__from(X)

where the Polynomial interpretation:
POL(n__from(x_1)) = x_1
POL(s(x_1)) = x_1
POL(from(x_1)) = 1 + 2*x_1
POL(cons(x_1, x_2)) = x_1 + x_2
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.608 seconds.