Term Rewriting System R:

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

Termination of R to be shown.


R contains the following Dependency Pairs: 


   P(0) -> G(0)
   F(g(x), g(y)) -> F(p(f(g(x), s(y))), g(s(p(x))))
   F(g(x), g(y)) -> P(f(g(x), s(y)))
   F(g(x), g(y)) -> F(g(x), s(y))
   F(g(x), g(y)) -> G(s(p(x)))
   F(g(x), g(y)) -> P(x)

Furthermore, R contains one SCC.


SCC1:

F(g(x), g(y)) -> F(p(f(g(x), s(y))), g(s(p(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: 

   f(g(x), g(y)) -> f(p(f(g(x), s(y))), g(s(p(x))))
   g(s(p(x))) -> p(x)
   p(0) -> g(0)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(g(x_1)) = 1 + x_1
POL(s(x_1)) = x_1
POL(F(x_1, x_2)) = x_1*x_2
POL(f(x_1, x_2)) = 0
POL(0) = 1
POL(p(x_1)) = 2*x_1

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 24.54 seconds.