Term Rewriting System R:

   [X, Y, Z, X1, X2]
   active(2nd(cons(X, cons(Y, Z)))) -> mark(Y)
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(2nd(X)) -> 2nd(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   2nd(mark(X)) -> mark(2nd(X))
   2nd(ok(X)) -> ok(2nd(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   from(ok(X)) -> ok(from(X))
   s(mark(X)) -> mark(s(X))
   s(ok(X)) -> ok(s(X))
   proper(2nd(X)) -> 2nd(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Termination of R to be shown.


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

   active(2nd(cons(X, cons(Y, Z)))) -> mark(Y)
   active(2nd(X)) -> 2nd(active(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   top(ok(X)) -> top(active(X))

where the Polynomial interpretation:
POL(active(x_1)) = 2*x_1
POL(proper(x_1)) = x_1
POL(s(x_1)) = x_1
POL(top(x_1)) = 1 + x_1
POL(2nd(x_1)) = 1 + 2*x_1
POL(mark(x_1)) = x_1
POL(from(x_1)) = x_1
POL(ok(x_1)) = 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: 

   active(from(X)) -> mark(cons(X, from(s(X))))

where the Polynomial interpretation:
POL(active(x_1)) = 1 + 2*x_1
POL(proper(x_1)) = x_1
POL(s(x_1)) = x_1
POL(top(x_1)) = 1 + x_1
POL(2nd(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(from(x_1)) = x_1
POL(ok(x_1)) = 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: 

   active(cons(X1, X2)) -> cons(active(X1), X2)

where the Polynomial interpretation:
POL(active(x_1)) = 2*x_1
POL(proper(x_1)) = x_1
POL(s(x_1)) = x_1
POL(top(x_1)) = 2 + x_1
POL(2nd(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(from(x_1)) = x_1
POL(ok(x_1)) = x_1
POL(cons(x_1, x_2)) = 1 + 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: 

   active(from(X)) -> from(active(X))

where the Polynomial interpretation:
POL(active(x_1)) = 2*x_1
POL(proper(x_1)) = x_1
POL(s(x_1)) = x_1
POL(top(x_1)) = 2 + x_1
POL(2nd(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(from(x_1)) = 1 + x_1
POL(ok(x_1)) = 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: 

   active(s(X)) -> s(active(X))

where the Polynomial interpretation:
POL(active(x_1)) = 2*x_1
POL(proper(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(top(x_1)) = 2 + x_1
POL(2nd(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(from(x_1)) = x_1
POL(ok(x_1)) = 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: 

   s(mark(X)) -> mark(s(X))
   top(mark(X)) -> top(proper(X))

where the Polynomial interpretation:
POL(proper(x_1)) = x_1
POL(s(x_1)) = 2*x_1
POL(top(x_1)) = 2 + x_1
POL(2nd(x_1)) = x_1
POL(mark(x_1)) = 1 + x_1
POL(from(x_1)) = x_1
POL(ok(x_1)) = 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: 

   s(ok(X)) -> ok(s(X))

where the Polynomial interpretation:
POL(proper(x_1)) = x_1
POL(s(x_1)) = 2*x_1
POL(2nd(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(from(x_1)) = x_1
POL(ok(x_1)) = 1 + 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(mark(X)) -> mark(from(X))

where the Polynomial interpretation:
POL(proper(x_1)) = x_1
POL(s(x_1)) = x_1
POL(2nd(x_1)) = x_1
POL(mark(x_1)) = 1 + x_1
POL(from(x_1)) = 2*x_1
POL(ok(x_1)) = 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(ok(X)) -> ok(from(X))

where the Polynomial interpretation:
POL(proper(x_1)) = x_1
POL(s(x_1)) = x_1
POL(2nd(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(from(x_1)) = 2*x_1
POL(ok(x_1)) = 1 + 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: 

   cons(mark(X1), X2) -> mark(cons(X1, X2))

where the Polynomial interpretation:
POL(proper(x_1)) = x_1
POL(s(x_1)) = x_1
POL(2nd(x_1)) = x_1
POL(mark(x_1)) = 1 + x_1
POL(from(x_1)) = x_1
POL(ok(x_1)) = x_1
POL(cons(x_1, x_2)) = 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: 

   2nd(mark(X)) -> mark(2nd(X))

where the Polynomial interpretation:
POL(proper(x_1)) = x_1
POL(s(x_1)) = x_1
POL(2nd(x_1)) = 2*x_1
POL(mark(x_1)) = 1 + x_1
POL(from(x_1)) = x_1
POL(ok(x_1)) = 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: 

   2nd(ok(X)) -> ok(2nd(X))

where the Polynomial interpretation:
POL(proper(x_1)) = x_1
POL(s(x_1)) = x_1
POL(2nd(x_1)) = 2*x_1
POL(from(x_1)) = x_1
POL(ok(x_1)) = 1 + 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: 

   proper(2nd(X)) -> 2nd(proper(X))

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

   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))

where the Polynomial interpretation:
POL(proper(x_1)) = 2*x_1
POL(s(x_1)) = x_1
POL(from(x_1)) = x_1
POL(cons(x_1, x_2)) = 1 + 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: 

   proper(from(X)) -> from(proper(X))

where the Polynomial interpretation:
POL(proper(x_1)) = 2*x_1
POL(s(x_1)) = x_1
POL(from(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: 

   proper(s(X)) -> s(proper(X))

where the Polynomial interpretation:
POL(proper(x_1)) = 2*x_1
POL(s(x_1)) = 1 + x_1
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.659 seconds.