Term Rewriting System R:

   [x, y, u, v]
   s(a) -> a
   s(s(x)) -> x
   s(f(x, y)) -> f(s(y), s(x))
   s(g(x, y)) -> g(s(x), s(y))
   f(x, a) -> x
   f(a, y) -> y
   f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
   g(a, a) -> a

Termination of R to be shown.


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

   s(a) -> a
   f(x, a) -> x
   f(a, y) -> y
   g(a, a) -> a

where the Polynomial interpretation:
POL(g(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = 2*x_1
POL(a) = 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: 

   s(s(x)) -> x

where the Polynomial interpretation:
POL(g(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = 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: 

   s(f(x, y)) -> f(s(y), s(x))
   s(g(x, y)) -> g(s(x), s(y))

where the Polynomial interpretation:
POL(g(x_1, x_2)) = 1 + x_1 + x_2
POL(s(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: 

   f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))

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



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.437 seconds.