Term Rewriting System R:

   [X, Y, X1, X2]
   active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
   active(f(X1, X2)) -> f(active(X1), X2)
   active(g(X)) -> g(active(X))
   f(mark(X1), X2) -> mark(f(X1, X2))
   f(ok(X1), ok(X2)) -> ok(f(X1, X2))
   g(mark(X)) -> mark(g(X))
   g(ok(X)) -> ok(g(X))
   proper(f(X1, X2)) -> f(proper(X1), proper(X2))
   proper(g(X)) -> g(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(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
   f(ok(X1), ok(X2)) -> ok(f(X1, X2))

where the Polynomial interpretation:
POL(active(x_1)) = 1 + 2*x_1
POL(g(x_1)) = x_1
POL(proper(x_1)) = x_1
POL(top(x_1)) = 1 + x_1
POL(mark(x_1)) = x_1
POL(ok(x_1)) = 1 + 2*x_1
POL(f(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(f(X1, X2)) -> f(active(X1), X2)

where the Polynomial interpretation:
POL(active(x_1)) = 2*x_1
POL(g(x_1)) = x_1
POL(proper(x_1)) = x_1
POL(top(x_1)) = 2 + x_1
POL(mark(x_1)) = x_1
POL(ok(x_1)) = 2*x_1
POL(f(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(g(X)) -> g(active(X))
   top(ok(X)) -> top(active(X))

where the Polynomial interpretation:
POL(g(x_1)) = 1 + 2*x_1
POL(active(x_1)) = 2*x_1
POL(proper(x_1)) = x_1
POL(top(x_1)) = 2 + x_1
POL(mark(x_1)) = x_1
POL(ok(x_1)) = 1 + 2*x_1
POL(f(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: 

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

where the Polynomial interpretation:
POL(g(x_1)) = 2*x_1
POL(proper(x_1)) = x_1
POL(top(x_1)) = 2 + x_1
POL(mark(x_1)) = 1 + x_1
POL(ok(x_1)) = x_1
POL(f(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: 

   g(ok(X)) -> ok(g(X))

where the Polynomial interpretation:
POL(g(x_1)) = 2*x_1
POL(proper(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(ok(x_1)) = 1 + x_1
POL(f(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: 

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

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(proper(x_1)) = x_1
POL(mark(x_1)) = 1 + x_1
POL(f(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: 

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

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(proper(x_1)) = 2*x_1
POL(f(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(g(X)) -> g(proper(X))

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



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.511 seconds.