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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


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


Dependency Pairs:

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


Rule:


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





The following dependency pair can be strictly oriented:

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


The following usable rule using the Ce-refinement can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(a)=  1  
  POL(F(x1, x2))=  x1 + x2  
  POL(f)=  0  

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


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




The following remains to be proven:
Dependency Pair:

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


Rule:


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




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