Term Rewriting System R:

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

Termination of R to be shown.


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


R contains the following Dependency Pairs: 


   F(g(x), a) -> F(x, g(a))
   F(g(x), g(y)) -> H(g(y), x, g(y))
   H(g(x), y, z) -> F(y, h(x, y, z))
   H(g(x), y, z) -> H(x, y, z)

Furthermore, R contains one SCC.


SCC1:

H(g(x), y, z) -> H(x, y, z)
H(g(x), y, z) -> F(y, h(x, y, z))
F(g(x), g(y)) -> H(g(y), x, g(y))
F(g(x), a) -> F(x, g(a))

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

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(g(x_1)) = 1 + x_1
POL(a) = 0
POL(h(x_1, x_2, x_3)) = 0
POL(F(x_1, x_2)) = x_1
POL(f(x_1, x_2)) = 0
POL(H(x_1, x_2, x_3)) = 1 + x_2

 resulting in one subcycle.


SCC1.Polo1:

H(g(x), y, z) -> H(x, y, z)

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, x_2, x_3)) = 1 + x_1 + x_2 + x_3

The following Dependency Pairs can be deleted:

   H(g(x), y, z) -> H(x, y, z)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.717 seconds.