Term Rewriting System R:

   [X, Y, Z, X1, X2]
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> cons(X, n__from(s(X)))
   from(X) -> n__from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(X) -> 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 + 2*x_2
POL(n__first(x_1, x_2)) = x_1 + x_2
POL(n__from(x_1)) = x_1
POL(nil) = 0
POL(s(x_1)) = x_1
POL(activate(x_1)) = 2*x_1
POL(from(x_1)) = 2*x_1
POL(0) = 1
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: 

   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> n__from(X)

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

   first(X1, X2) -> n__first(X1, X2)

where the Polynomial interpretation:
POL(n__first(x_1, x_2)) = 1 + x_1 + x_2
POL(first(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(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__first(X1, X2)) -> first(X1, X2)

where the Polynomial interpretation:
POL(n__from(x_1)) = x_1
POL(n__first(x_1, x_2)) = 1 + x_1 + x_2
POL(first(x_1, x_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(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)
   activate(X) -> X

where the Polynomial interpretation:
POL(n__from(x_1)) = x_1
POL(activate(x_1)) = 1 + 2*x_1
POL(s(x_1)) = x_1
POL(from(x_1)) = 2*x_1
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)))

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: 0.467 seconds.