Term Rewriting System R:

   [x, y, z]
   f(0, 1, g(x, y), z) -> f(g(x, y), g(x, y), g(x, y), h(x))
   g(0, 1) -> 0
   g(0, 1) -> 1
   h(g(x, y)) -> h(x)

Termination of R to be shown.


R contains the following Dependency Pairs: 


   H(g(x, y)) -> H(x)
   F(0, 1, g(x, y), z) -> F(g(x, y), g(x, y), g(x, y), h(x))
   F(0, 1, g(x, y), z) -> H(x)

Furthermore, R contains two SCCs.


SCC1:

H(g(x, y)) -> H(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, x_2)) = 1 + x_1 + x_2
POL(H(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

F(0, 1, g(x, y), z) -> F(g(x, y), g(x, y), g(x, y), h(x))

Found an infinite P-chain over R:
P = F(0, 1, g(x, y), z) -> F(g(x, y), g(x, y), g(x, y), h(x))
R = [h(g(x, y)) -> h(x), g(0, 1) -> 1, g(0, 1) -> 0, f(0, 1, g(x, y), z) -> f(g(x, y), g(x, y), g(x, y), h(x))]
s = F(0, 1, g(0, 1), h(0)) evaluates to t = F(0, 1, g(0, 1), h(0))
Thus, s starts an infinite reduction.


Non-Termination of R could be shown.

Duration: 0.652 seconds.