Term Rewriting System R:

   [x, y]
   g(0, f(x, x)) -> x
   g(x, s(y)) -> g(f(x, y), 0)
   g(s(x), y) -> g(f(x, y), 0)
   g(f(x, y), 0) -> f(g(x, 0), g(y, 0))

Termination of R to be shown.


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

   g(0, f(x, x)) -> x

where the Polynomial interpretation:
POL(g(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = 1 + x_1
POL(f(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
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(x, s(y)) -> g(f(x, y), 0)
   g(s(x), y) -> g(f(x, y), 0)

where the Polynomial interpretation:
POL(g(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = 1 + x_1
POL(f(x_1, x_2)) = x_1 + x_2
POL(0) = 0
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(f(x, y), 0) -> f(g(x, 0), g(y, 0))

where the Polynomial interpretation:
POL(g(x_1, x_2)) = 2*x_1 + x_2
POL(f(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.424 seconds.