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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rule:


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





The following dependency pair can be strictly oriented:

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


The following usable rule w.r.t. to the AFS can be oriented:

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
a > f

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> x1
f(x1, x2) -> f


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rule:


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




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