Term Rewriting System R:

   [x, y]
   f(x, y) -> g1(x, x, y)
   f(x, y) -> g1(y, x, x)
   f(x, y) -> g2(x, y, y)
   f(x, y) -> g2(y, y, x)
   g1(x, x, y) -> h(x, y)
   g1(y, x, x) -> h(x, y)
   g2(x, y, y) -> h(x, y)
   g2(y, y, x) -> h(x, y)
   h(x, x) -> x

Termination of R to be shown.


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

   f(x, y) -> g1(x, x, y)
   f(x, y) -> g1(y, x, x)
   f(x, y) -> g2(x, y, y)
   f(x, y) -> g2(y, y, x)

where the Polynomial interpretation:
POL(g1(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(h(x_1, x_2)) = x_1 + x_2
POL(f(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
POL(g2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
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: 

   g1(x, x, y) -> h(x, y)
   g1(y, x, x) -> h(x, y)

where the Polynomial interpretation:
POL(g1(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
POL(h(x_1, x_2)) = x_1 + x_2
POL(g2(x_1, x_2, x_3)) = x_1 + x_2 + x_3
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: 

   h(x, x) -> x

where the Polynomial interpretation:
POL(h(x_1, x_2)) = 1 + x_1 + x_2
POL(g2(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
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: 

   g2(x, y, y) -> h(x, y)
   g2(y, y, x) -> h(x, y)

where the Polynomial interpretation:
POL(h(x_1, x_2)) = x_1 + x_2
POL(g2(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.427 seconds.