Term Rewriting System R:
[x]
h(f(f(x))) -> h(f(g(f(x))))
f(g(f(x))) -> f(f(x))

Innermost Termination of R to be shown.



   TRS
Dependency Pair Analysis
Rev
Rev



R contains the following Dependency Pairs:

H(f(f(x))) -> H(f(g(f(x))))
H(f(f(x))) -> F(g(f(x)))
F(g(f(x))) -> F(f(x))

Furthermore, R contains one SCC.


   TRS
DPs
       →DP Problem 1
Non Termination
Rev
Rev


Dependency Pair:

H(f(f(x))) -> H(f(g(f(x))))


Rules:


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


Strategy:

innermost




Found an infinite P-chain over R:
P =

H(f(f(x))) -> H(f(g(f(x))))

R =

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

s = H(f(g(f(x'''))))
evaluates to t =H(f(g(f(x'''))))

Thus, s starts an infinite chain.




Trying another alternative:
   TRS
DPs
Reversing
Rev




Rule(s) before reversing:

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


Rule(s) after reversing:

f'(f'(h'(x))) -> f'(g'(f'(h'(x))))
f'(g'(f'(x))) -> f'(f'(x))





Trying another alternative:
   TRS
DPs
Rev
Reversing




Rule(s) before reversing:

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


Rule(s) after reversing:

f'(f'(h'(x))) -> f'(g'(f'(h'(x))))
f'(g'(f'(x))) -> f'(f'(x))


Innermost Termination of R could not be shown.
Duration:
0:02 minutes