Term Rewriting System R:

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

where the Polynomial interpretation:
POL(g1(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
POL(a) = 0
POL(h(x_1, x_2)) = x_1 + x_2
POL(f(x_1, x_2, x_3, x_4, x_5)) = 1 + 2*x_1 + 2*x_2 + x_3 + x_4 + x_5
POL(g2(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
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, a) -> h(x, y)
   g1(y, x, x, a) -> h(x, y)
   g2(x, y, y, a) -> h(x, y)
   g2(y, y, x, a) -> h(x, y)

where the Polynomial interpretation:
POL(g1(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
POL(a) = 1
POL(h(x_1, x_2)) = x_1 + x_2
POL(g2(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
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
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.447 seconds.