Term Rewriting System R:

   [x, y]
   minus(minus(x)) -> x
   minus(h(x)) -> h(minus(x))
   minus(f(x, y)) -> f(minus(y), minus(x))

Termination of R to be shown.


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

   minus(minus(x)) -> x

where the Polynomial interpretation:
POL(minus(x_1)) = 1 + 2*x_1
POL(h(x_1)) = 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: 

   minus(h(x)) -> h(minus(x))

where the Polynomial interpretation:
POL(minus(x_1)) = 2*x_1
POL(h(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: 

   minus(f(x, y)) -> f(minus(y), minus(x))

where the Polynomial interpretation:
POL(minus(x_1)) = 2*x_1
POL(f(x_1, x_2)) = 1 + x_1 + x_2
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.397 seconds.