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))))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



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.


   R
DPs
       →DP Problem 1
Non Termination


Dependency Pairs:

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))


Rule:


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


Strategy:

innermost




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(g(f(x'', y'')))
evaluates to t =G(g(f(g(g(x'')), g(g(y'')))))

Thus, s starts an infinite chain as s matches t.

Innermost Non-Termination of R could be shown.
Duration:
0:00 minutes