Term Rewriting System R:

   [x, y]
   or(x, x) -> x
   and(x, x) -> x
   not(not(x)) -> x
   not(and(x, y)) -> or(not(x), not(y))
   not(or(x, y)) -> and(not(x), not(y))

Termination of R to be shown.


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

   or(x, x) -> x
   and(x, x) -> x

where the Polynomial interpretation:
POL(not(x_1)) = x_1
POL(and(x_1, x_2)) = 1 + x_1 + x_2
POL(or(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: 

   not(not(x)) -> x

where the Polynomial interpretation:
POL(not(x_1)) = 1 + 2*x_1
POL(and(x_1, x_2)) = 1 + x_1 + x_2
POL(or(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: 

   not(and(x, y)) -> or(not(x), not(y))
   not(or(x, y)) -> and(not(x), not(y))

where the Polynomial interpretation:
POL(not(x_1)) = 2*x_1
POL(or(x_1, x_2)) = 1 + x_1 + x_2
POL(and(x_1, x_2)) = 1 + x_1 + x_2
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.416 seconds.