Term Rewriting System R:

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

Termination of R to be shown.


R contains the following Dependency Pairs: 


   F(a, h(x)) -> F(g(x), h(x))
   F(a, h(x)) -> G(x)
   H(g(x)) -> H(a)
   G(h(x)) -> G(x)

Furthermore, R contains two SCCs.


SCC1:

G(h(x)) -> G(x)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(G(x_1)) = 1 + x_1
POL(h(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   G(h(x)) -> G(x)



 This transformation is resulting in no new subcycles.


SCC2:

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

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

Additionally, the following rule can be oriented: 

   g(h(x)) -> g(x)


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

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.482 seconds.