Term Rewriting System R:

   [x, y]
   f(x, y) -> x
   g(a) -> h(a, b, a)
   i(x) -> f(x, x)
   h(x, x, y) -> g(x)

Termination of R to be shown.


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

   f(x, y) -> x

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(b) = 0
POL(i(x_1)) = 1 + 2*x_1
POL(a) = 0
POL(h(x_1, x_2, x_3)) = x_1 + x_2 + x_3
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: 

   i(x) -> f(x, x)

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(b) = 0
POL(i(x_1)) = 1 + 2*x_1
POL(a) = 0
POL(h(x_1, x_2, x_3)) = x_1 + x_2 + x_3
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.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   H(x, x, y) -> G(x)
   G(a) -> H(a, b, a)

R contains no SCCs.




Termination of R successfully shown.


Duration: 0.446 seconds.