Term Rewriting System R:

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

Termination of R to be shown.


R contains the following Dependency Pairs: 


   H(x, x) -> G(x, 0)
   F(f(x)) -> F(h(f(x), f(x)))
   F(f(x)) -> H(f(x), f(x))
   F(f(x)) -> F(g(f(x), x))
   F(f(x)) -> G(f(x), x)

Furthermore, R contains one SCC.


SCC1:

F(f(x)) -> F(g(f(x), x))
F(f(x)) -> F(h(f(x), f(x)))

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   h(x, x) -> g(x, 0)
   g(x, y) -> y


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(g(x_1, x_2)) = x_2
POL(h(x_1, x_2)) = 0
POL(F(x_1)) = 1 + x_1
POL(f(x_1)) = 1 + x_1
POL(0) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.550 seconds.