Term Rewriting System R:
[y, x]
f(y, f(x, f(a, x))) -> f(f(f(a, x), f(x, a)), f(a, y))
f(x, f(x, y)) -> f(f(f(x, a), a), a)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(y, f(x, f(a, x))) -> F(f(f(a, x), f(x, a)), f(a, y))
F(y, f(x, f(a, x))) -> F(f(a, x), f(x, a))
F(y, f(x, f(a, x))) -> F(x, a)
F(y, f(x, f(a, x))) -> F(a, y)
F(x, f(x, y)) -> F(f(f(x, a), a), a)
F(x, f(x, y)) -> F(f(x, a), a)
F(x, f(x, y)) -> F(x, a)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

F(y, f(x, f(a, x))) -> F(a, y)
F(y, f(x, f(a, x))) -> F(f(a, x), f(x, a))
F(y, f(x, f(a, x))) -> F(f(f(a, x), f(x, a)), f(a, y))


Rules:


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




Termination of R could not be shown.
Duration:
0:00 minutes