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