Term Rewriting System R:

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

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: 


   G(f(x, y)) -> G(g(x))
   G(f(x, y)) -> G(x)
   G(f(x, y)) -> G(g(y))
   G(f(x, y)) -> G(y)

Furthermore, R contains one SCC.


SCC1:

G(f(x, y)) -> G(y)
G(f(x, y)) -> G(g(y))
G(f(x, y)) -> G(x)
G(f(x, y)) -> G(g(x))

Found an infinite P-chain over R:
P = G(f(x, y)) -> G(y)
G(f(x, y)) -> G(g(y))
G(f(x, y)) -> G(x)
G(f(x, y)) -> G(g(x))
R = [g(f(x, y)) -> f(f(g(g(x)), g(g(y))), f(g(g(x)), g(g(y))))]
s = G(f(x, f(x'', y''))) evaluates to t = G(f(f(g(g(x'')), g(g(y''))), f(g(g(x'')), g(g(y'')))))
Thus, s starts an infinite reduction as s matches t.


Non-Termination of R could be shown.

Duration: 0.665 seconds.